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IncompressibleNavierStokes.jl

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Citing ​

See the CITATION.cff file in the root of the repository for citation information. To get the citation in BibTeX format, click on "Cite this repository"-button to the right on the GitHub landing page.

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IncompressibleNavierStokes.jl

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Contributing ​

If you encounter errors, typos, or want to propose additional features, feel free to open issues or pull request at the GitHub repository.

- + \ No newline at end of file diff --git a/dev/about/development.html b/dev/about/development.html index 4a24cf0af..7e2b17cb7 100644 --- a/dev/about/development.html +++ b/dev/about/development.html @@ -9,9 +9,9 @@ - + - + @@ -25,7 +25,7 @@
IncompressibleNavierStokes.jl

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Local development ​

Use Juliaup ​

Install Julia using the juliaup version manager. This allows for choosing the Julia version, e.g. v1.11, by running

juliaup add 1.11
 juliaup default 1.11

Revise ​

It is recommended to use Revise.jl for interactive development. Add it to your global environment with

julia -e 'using Pkg; Pkg.add("Revise")'

and load it in the startup file (create the file and folder if it is not already there) at ~/.julia/config/startup.jl with

using Revise

Then changes to the IncompressibleNavierStokes modules are detected and reloaded live.

Environments ​

To keep dependencies sparse, there are multiple Project.toml files in this repository, specifying environments. For example, the docs environment contains packages that are required to build documentation, but not needed to run the simulations. To add local packages to an environment and be detectable by Revise, they need to be Pkg.developed. For example, the package NeuralClosure depends on IncompressibleNavierStokes, and IncompressibleNavierStokes needs to be deved with

julia --project=lib/NeuralClosure -e 'using Pkg; Pkg.develop(PackageSpec(; path = "."))'

Run this from the repository root, where "." is the path to IncompressibleNavierStokes.

On Julia v1.11, this linking is automatic, with the dedicated [sources] sections in the Project.toml files. In that case, an environment can be instantiated with

julia --project=lib/NeuralClosure -e 'using Pkg; Pkg.instantiate()'

etc., or interactively from the REPL with ] instantiate.

VSCode ​

In VSCode, you can choose an active environment by clicking on the Julia env: button in the status bar, or press ctrl/cmd + shift + p and start typing environment:

  • Julia: Activate this environment activates the one of the current open file

  • Julia: Change current environment otherwise

Then scripts will be run from the selected environment.

Environment vs package ​

If a Project.toml has a header with a name and uuid, then it is a package with the module src/ModuleName.jl, and can be depended on in other projects (by add or dev).

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IncompressibleNavierStokes.jl

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IncompressibleNavierStokes.jl

Appearance

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IncompressibleNavierStokes.jl

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License ​

IncompressibleNavierStokes.license Constant

MIT License

Copyright (c) 2021 Syver Døving Agdestein, Benjamin Sanderse, and contributors

Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:

The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.

THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.

source

- +
IncompressibleNavierStokes.jl

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License ​

IncompressibleNavierStokes.license Constant

MIT License

Copyright (c) 2021 Syver Døving Agdestein, Benjamin Sanderse, and contributors

Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:

The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.

THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.

source

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Energy-conserving solvers for the incompressible Navier-Stokes equations.

Exports

The following symbols are exported by IncompressibleNavierStokes:

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Energy-conserving solvers for the incompressible Navier-Stokes equations.

Exports

The following symbols are exported by IncompressibleNavierStokes:

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Energy-conserving solvers for the incompressible Navier-Stokes equations.

Exports

The following symbols are exported by IncompressibleNavierStokes:

source

',5))])])}const f=r(c,[["render",m]]);export{I as __pageData,f as default}; +import{_ as r,c as i,j as s,a as o,G as a,a5 as p,B as n,o as t}from"./chunks/framework.Dr1kDDE8.js";const I=JSON.parse('{"title":"About IncompressibleNavierStokes.jl","description":"","frontmatter":{},"headers":[],"relativePath":"about/index.md","filePath":"about/index.md","lastUpdated":null}'),c={name:"about/index.md"},d={class:"jldocstring custom-block",open:""};function m(v,e,u,b,k,N){const l=n("Badge");return t(),i("div",null,[e[3]||(e[3]=s("h1",{id:"About-IncompressibleNavierStokes.jl",tabindex:"-1"},[o("About IncompressibleNavierStokes.jl "),s("a",{class:"header-anchor",href:"#About-IncompressibleNavierStokes.jl","aria-label":'Permalink to "About IncompressibleNavierStokes.jl {#About-IncompressibleNavierStokes.jl}"'},"​")],-1)),s("details",d,[s("summary",null,[e[0]||(e[0]=s("a",{id:"IncompressibleNavierStokes.IncompressibleNavierStokes",href:"#IncompressibleNavierStokes.IncompressibleNavierStokes"},[s("span",{class:"jlbinding"},"IncompressibleNavierStokes.IncompressibleNavierStokes")],-1)),e[1]||(e[1]=o()),a(l,{type:"info",class:"jlObjectType jlModule",text:"Module"})]),e[2]||(e[2]=p('

Energy-conserving solvers for the incompressible Navier-Stokes equations.

Exports

The following symbols are exported by IncompressibleNavierStokes:

source

',5))])])}const f=r(c,[["render",m]]);export{I as __pageData,f as default}; diff --git a/dev/assets/about_license.md.BMTPbAIg.js b/dev/assets/about_license.md.B43Z_-HQ.js similarity index 92% rename from dev/assets/about_license.md.BMTPbAIg.js rename to dev/assets/about_license.md.B43Z_-HQ.js index 64ea2b795..053394c3a 100644 --- a/dev/assets/about_license.md.BMTPbAIg.js +++ b/dev/assets/about_license.md.B43Z_-HQ.js @@ -1 +1 @@ -import{_ as n,c as i,j as t,a as s,G as r,B as a,o as l}from"./chunks/framework.Dr1kDDE8.js";const u=JSON.parse('{"title":"License","description":"","frontmatter":{},"headers":[],"relativePath":"about/license.md","filePath":"about/license.md","lastUpdated":null}'),d={name:"about/license.md"},p={class:"jldocstring custom-block",open:""};function T(I,e,N,O,c,E){const o=a("Badge");return l(),i("div",null,[e[8]||(e[8]=t("h1",{id:"license",tabindex:"-1"},[s("License "),t("a",{class:"header-anchor",href:"#license","aria-label":'Permalink to "License"'},"​")],-1)),t("details",p,[t("summary",null,[e[0]||(e[0]=t("a",{id:"IncompressibleNavierStokes.license",href:"#IncompressibleNavierStokes.license"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.license")],-1)),e[1]||(e[1]=s()),r(o,{type:"info",class:"jlObjectType jlConstant",text:"Constant"})]),e[2]||(e[2]=t("p",null,"MIT License",-1)),e[3]||(e[3]=t("p",null,"Copyright (c) 2021 Syver Døving Agdestein, Benjamin Sanderse, and contributors",-1)),e[4]||(e[4]=t("p",null,'Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:',-1)),e[5]||(e[5]=t("p",null,"The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.",-1)),e[6]||(e[6]=t("p",null,'THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. 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IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.',-1)),e[7]||(e[7]=t("p",null,[t("a",{href:"https://github.com/agdestein/IncompressibleNavierStokes.jl/blob/03acafa3566742b1f37d722d2207f5dc21d031c0/src/IncompressibleNavierStokes.jl#L56",target:"_blank",rel:"noreferrer"},"source")],-1))])])}const S=n(d,[["render",T]]);export{u as __pageData,S as default}; +import{_ as n,c as i,j as t,a as s,G as r,B as a,o as l}from"./chunks/framework.Dr1kDDE8.js";const u=JSON.parse('{"title":"License","description":"","frontmatter":{},"headers":[],"relativePath":"about/license.md","filePath":"about/license.md","lastUpdated":null}'),d={name:"about/license.md"},p={class:"jldocstring custom-block",open:""};function T(I,e,N,O,E,c){const o=a("Badge");return l(),i("div",null,[e[8]||(e[8]=t("h1",{id:"license",tabindex:"-1"},[s("License "),t("a",{class:"header-anchor",href:"#license","aria-label":'Permalink to "License"'},"​")],-1)),t("details",p,[t("summary",null,[e[0]||(e[0]=t("a",{id:"IncompressibleNavierStokes.license",href:"#IncompressibleNavierStokes.license"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.license")],-1)),e[1]||(e[1]=s()),r(o,{type:"info",class:"jlObjectType jlConstant",text:"Constant"})]),e[2]||(e[2]=t("p",null,"MIT License",-1)),e[3]||(e[3]=t("p",null,"Copyright (c) 2021 Syver Døving Agdestein, Benjamin Sanderse, and contributors",-1)),e[4]||(e[4]=t("p",null,'Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:',-1)),e[5]||(e[5]=t("p",null,"The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.",-1)),e[6]||(e[6]=t("p",null,'THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.',-1)),e[7]||(e[7]=t("p",null,[t("a",{href:"https://github.com/agdestein/IncompressibleNavierStokes.jl/blob/a63d20e3a410f14ba6cc612a27e89f8974019d9f/src/IncompressibleNavierStokes.jl#L56",target:"_blank",rel:"noreferrer"},"source")],-1))])])}const S=n(d,[["render",T]]);export{u as __pageData,S as default}; diff --git a/dev/assets/about_versions.md.ByqO0kAn.js b/dev/assets/about_versions.md.Bk-sITSg.js similarity index 97% rename from dev/assets/about_versions.md.ByqO0kAn.js rename to dev/assets/about_versions.md.Bk-sITSg.js index bbf670922..f5a53a5a8 100644 --- a/dev/assets/about_versions.md.ByqO0kAn.js +++ b/dev/assets/about_versions.md.Bk-sITSg.js @@ -21,7 +21,7 @@ import{_ as n,c as a,a5 as p,o as l}from"./chunks/framework.Dr1kDDE8.js";const b [27a7e980] Animations v0.4.2 [dce04be8] ArgCheck v2.4.0 [4fba245c] ArrayInterface v7.17.1 - [a9b6321e] Atomix v1.0.0 + [a9b6321e] Atomix v1.0.1 [67c07d97] Automa v1.1.0 [13072b0f] AxisAlgorithms v1.1.0 [39de3d68] AxisArrays v0.4.7 @@ -34,7 +34,7 @@ import{_ as n,c as a,a5 as p,o as l}from"./chunks/framework.Dr1kDDE8.js";const b [fa961155] CEnum v0.5.0 [2a0fbf3d] CPUSummary v0.2.6 [159f3aea] Cairo v1.1.1 - [13f3f980] CairoMakie v0.12.17 + [13f3f980] CairoMakie v0.12.18 [082447d4] ChainRules v1.72.1 [d360d2e6] ChainRulesCore v1.25.0 [fb6a15b2] CloseOpenIntervals v0.1.13 @@ -71,7 +71,7 @@ import{_ as n,c as a,a5 as p,o as l}from"./chunks/framework.Dr1kDDE8.js";const b [daee34ce] DocumenterCitations v1.3.5 [4710194d] DocumenterVitepress v0.1.3 [4e289a0a] EnumX v1.0.4 - [7da242da] Enzyme v0.13.21 + [7da242da] Enzyme v0.13.22 [f151be2c] EnzymeCore v0.8.8 [429591f6] ExactPredicates v2.2.8 [318dbb63] Examples v1.0.0 \`../examples\` @@ -97,7 +97,7 @@ import{_ as n,c as a,a5 as p,o as l}from"./chunks/framework.Dr1kDDE8.js";const b [77dc65aa] FunctionWrappersWrappers v0.1.3 [d9f16b24] Functors v0.5.2 [f7f18e0c] GLFW v3.4.3 - [e9467ef8] GLMakie v0.10.17 + [e9467ef8] GLMakie v0.10.18 ⌃ [0c68f7d7] GPUArrays v10.3.1 ⌅ [46192b85] GPUArraysCore v0.1.6 [61eb1bfa] GPUCompiler v1.0.1 @@ -138,7 +138,7 @@ import{_ as n,c as a,a5 as p,o as l}from"./chunks/framework.Dr1kDDE8.js";const b [7d188eb4] JSONSchema v1.4.1 [b835a17e] JpegTurbo v0.1.5 [b14d175d] JuliaVariables v0.2.4 - [63c18a36] KernelAbstractions v0.9.30 + [63c18a36] KernelAbstractions v0.9.31 [5ab0869b] KernelDensity v0.6.9 [929cbde3] LLVM v9.1.3 [b964fa9f] LaTeXStrings v1.4.0 @@ -147,7 +147,7 @@ import{_ as n,c as a,a5 as p,o as l}from"./chunks/framework.Dr1kDDE8.js";const b [8cdb02fc] LazyModules v0.3.1 [9c8b4983] LightXML v0.9.1 [98b081ad] Literate v2.20.1 - [2ab3a3ac] LogExpFunctions v0.3.28 + [2ab3a3ac] LogExpFunctions v0.3.29 [b2108857] Lux v1.4.1 [bb33d45b] LuxCore v1.2.1 [82251201] LuxLib v1.3.10 @@ -155,8 +155,8 @@ import{_ as n,c as a,a5 as p,o as l}from"./chunks/framework.Dr1kDDE8.js";const b [d8e11817] MLStyle v0.4.17 [f1d291b0] MLUtils v0.4.4 [1914dd2f] MacroTools v0.5.13 - [ee78f7c6] Makie v0.21.17 - [20f20a25] MakieCore v0.8.11 + [ee78f7c6] Makie v0.21.18 + [20f20a25] MakieCore v0.8.12 [d125e4d3] ManualMemory v0.1.8 [dbb5928d] MappedArrays v0.4.2 [d0879d2d] MarkdownAST v0.1.2 @@ -176,7 +176,7 @@ import{_ as n,c as a,a5 as p,o as l}from"./chunks/framework.Dr1kDDE8.js";const b [510215fc] Observables v0.5.5 [6fe1bfb0] OffsetArrays v1.14.2 [52e1d378] OpenEXR v0.3.3 - [3bd65402] Optimisers v0.4.1 + [3bd65402] Optimisers v0.4.2 [bac558e1] OrderedCollections v1.7.0 [bbf590c4] OrdinaryDiffEqCore v1.13.0 [b1df2697] OrdinaryDiffEqTsit5 v1.1.0 @@ -227,7 +227,7 @@ import{_ as n,c as a,a5 as p,o as l}from"./chunks/framework.Dr1kDDE8.js";const b [45858cf5] Sixel v0.1.3 [a2af1166] SortingAlgorithms v1.2.1 [dc90abb0] SparseInverseSubset v0.1.2 - [276daf66] SpecialFunctions v2.4.0 + [276daf66] SpecialFunctions v2.5.0 [171d559e] SplittablesBase v0.1.15 [860ef19b] StableRNGs v1.0.2 [cae243ae] StackViews v0.1.1 @@ -237,7 +237,7 @@ import{_ as n,c as a,a5 as p,o as l}from"./chunks/framework.Dr1kDDE8.js";const b [1e83bf80] StaticArraysCore v1.4.3 [10745b16] Statistics v1.11.1 [82ae8749] StatsAPI v1.7.0 - [2913bbd2] StatsBase v0.34.3 + [2913bbd2] StatsBase v0.34.4 [4c63d2b9] StatsFuns v1.3.2 [7792a7ef] StrideArraysCore v0.5.7 [69024149] StringEncodings v0.3.7 @@ -260,8 +260,7 @@ import{_ as n,c as a,a5 as p,o as l}from"./chunks/framework.Dr1kDDE8.js";const b [3a884ed6] UnPack v1.0.2 [1cfade01] UnicodeFun v0.4.1 [1986cc42] Unitful v1.21.1 - [013be700] UnsafeAtomics v0.2.1 - [d80eeb9a] UnsafeAtomicsLLVM v0.2.2 + [013be700] UnsafeAtomics v0.3.0 [4004b06d] VTKBase v1.0.1 [e3aaa7dc] WebP v0.1.3 [d49dbf32] WeightInitializers v1.0.4 diff --git a/dev/assets/about_versions.md.ByqO0kAn.lean.js b/dev/assets/about_versions.md.Bk-sITSg.lean.js similarity index 97% rename from dev/assets/about_versions.md.ByqO0kAn.lean.js rename to dev/assets/about_versions.md.Bk-sITSg.lean.js index bbf670922..f5a53a5a8 100644 --- a/dev/assets/about_versions.md.ByqO0kAn.lean.js +++ b/dev/assets/about_versions.md.Bk-sITSg.lean.js @@ -21,7 +21,7 @@ import{_ as n,c as a,a5 as p,o as l}from"./chunks/framework.Dr1kDDE8.js";const b [27a7e980] Animations v0.4.2 [dce04be8] ArgCheck v2.4.0 [4fba245c] ArrayInterface v7.17.1 - [a9b6321e] Atomix v1.0.0 + [a9b6321e] Atomix v1.0.1 [67c07d97] Automa v1.1.0 [13072b0f] AxisAlgorithms v1.1.0 [39de3d68] AxisArrays v0.4.7 @@ -34,7 +34,7 @@ import{_ as n,c as a,a5 as p,o as l}from"./chunks/framework.Dr1kDDE8.js";const b [fa961155] CEnum v0.5.0 [2a0fbf3d] CPUSummary v0.2.6 [159f3aea] Cairo v1.1.1 - [13f3f980] CairoMakie v0.12.17 + [13f3f980] CairoMakie v0.12.18 [082447d4] ChainRules v1.72.1 [d360d2e6] ChainRulesCore v1.25.0 [fb6a15b2] CloseOpenIntervals v0.1.13 @@ -71,7 +71,7 @@ import{_ as n,c as a,a5 as p,o as l}from"./chunks/framework.Dr1kDDE8.js";const b [daee34ce] DocumenterCitations v1.3.5 [4710194d] DocumenterVitepress v0.1.3 [4e289a0a] EnumX v1.0.4 - [7da242da] Enzyme v0.13.21 + [7da242da] Enzyme v0.13.22 [f151be2c] EnzymeCore v0.8.8 [429591f6] ExactPredicates v2.2.8 [318dbb63] Examples v1.0.0 \`../examples\` @@ -97,7 +97,7 @@ import{_ as n,c as a,a5 as p,o as l}from"./chunks/framework.Dr1kDDE8.js";const b [77dc65aa] FunctionWrappersWrappers v0.1.3 [d9f16b24] Functors v0.5.2 [f7f18e0c] GLFW v3.4.3 - [e9467ef8] GLMakie v0.10.17 + [e9467ef8] GLMakie v0.10.18 ⌃ [0c68f7d7] GPUArrays v10.3.1 ⌅ [46192b85] GPUArraysCore v0.1.6 [61eb1bfa] GPUCompiler v1.0.1 @@ -138,7 +138,7 @@ import{_ as n,c as a,a5 as p,o as l}from"./chunks/framework.Dr1kDDE8.js";const b [7d188eb4] JSONSchema v1.4.1 [b835a17e] JpegTurbo v0.1.5 [b14d175d] JuliaVariables v0.2.4 - [63c18a36] KernelAbstractions v0.9.30 + [63c18a36] KernelAbstractions v0.9.31 [5ab0869b] KernelDensity v0.6.9 [929cbde3] LLVM v9.1.3 [b964fa9f] LaTeXStrings v1.4.0 @@ -147,7 +147,7 @@ import{_ as n,c as a,a5 as p,o as l}from"./chunks/framework.Dr1kDDE8.js";const b [8cdb02fc] LazyModules v0.3.1 [9c8b4983] LightXML v0.9.1 [98b081ad] Literate v2.20.1 - [2ab3a3ac] LogExpFunctions v0.3.28 + [2ab3a3ac] LogExpFunctions v0.3.29 [b2108857] Lux v1.4.1 [bb33d45b] LuxCore v1.2.1 [82251201] LuxLib v1.3.10 @@ -155,8 +155,8 @@ import{_ as n,c as a,a5 as p,o as l}from"./chunks/framework.Dr1kDDE8.js";const b [d8e11817] MLStyle v0.4.17 [f1d291b0] MLUtils v0.4.4 [1914dd2f] MacroTools v0.5.13 - [ee78f7c6] Makie v0.21.17 - [20f20a25] MakieCore v0.8.11 + [ee78f7c6] Makie v0.21.18 + [20f20a25] MakieCore v0.8.12 [d125e4d3] ManualMemory v0.1.8 [dbb5928d] MappedArrays v0.4.2 [d0879d2d] MarkdownAST v0.1.2 @@ -176,7 +176,7 @@ import{_ as n,c as a,a5 as p,o as l}from"./chunks/framework.Dr1kDDE8.js";const b [510215fc] Observables v0.5.5 [6fe1bfb0] OffsetArrays v1.14.2 [52e1d378] OpenEXR v0.3.3 - [3bd65402] Optimisers v0.4.1 + [3bd65402] Optimisers v0.4.2 [bac558e1] OrderedCollections v1.7.0 [bbf590c4] OrdinaryDiffEqCore v1.13.0 [b1df2697] OrdinaryDiffEqTsit5 v1.1.0 @@ -227,7 +227,7 @@ import{_ as n,c as a,a5 as p,o as l}from"./chunks/framework.Dr1kDDE8.js";const b [45858cf5] Sixel v0.1.3 [a2af1166] SortingAlgorithms v1.2.1 [dc90abb0] SparseInverseSubset v0.1.2 - [276daf66] SpecialFunctions v2.4.0 + [276daf66] SpecialFunctions v2.5.0 [171d559e] SplittablesBase v0.1.15 [860ef19b] StableRNGs v1.0.2 [cae243ae] StackViews v0.1.1 @@ -237,7 +237,7 @@ import{_ as n,c as a,a5 as p,o as l}from"./chunks/framework.Dr1kDDE8.js";const b [1e83bf80] StaticArraysCore v1.4.3 [10745b16] Statistics v1.11.1 [82ae8749] StatsAPI v1.7.0 - [2913bbd2] StatsBase v0.34.3 + [2913bbd2] StatsBase v0.34.4 [4c63d2b9] StatsFuns v1.3.2 [7792a7ef] StrideArraysCore v0.5.7 [69024149] StringEncodings v0.3.7 @@ -260,8 +260,7 @@ import{_ as n,c as a,a5 as p,o as l}from"./chunks/framework.Dr1kDDE8.js";const b [3a884ed6] UnPack v1.0.2 [1cfade01] UnicodeFun v0.4.1 [1986cc42] Unitful v1.21.1 - [013be700] UnsafeAtomics v0.2.1 - [d80eeb9a] UnsafeAtomicsLLVM v0.2.2 + [013be700] UnsafeAtomics v0.3.0 [4004b06d] VTKBase v1.0.1 [e3aaa7dc] WebP v0.1.3 [d49dbf32] WeightInitializers v1.0.4 diff --git a/dev/assets/app.D86iw_p-.js b/dev/assets/app.Cqn8wUEB.js similarity index 95% rename from dev/assets/app.D86iw_p-.js rename to dev/assets/app.Cqn8wUEB.js index 4547f450a..dc100c61f 100644 --- a/dev/assets/app.D86iw_p-.js +++ b/dev/assets/app.Cqn8wUEB.js @@ -1 +1 @@ -import{R as p}from"./chunks/theme.BCKAc1BQ.js";import{R as o,a6 as u,a7 as c,a8 as l,a9 as f,aa as d,ab as m,ac as h,ad as g,ae as A,af as v,d as P,u as R,v as w,s as y,ag as C,ah as b,ai as E,a4 as S}from"./chunks/framework.Dr1kDDE8.js";function i(e){if(e.extends){const a=i(e.extends);return{...a,...e,async enhanceApp(t){a.enhanceApp&&await a.enhanceApp(t),e.enhanceApp&&await e.enhanceApp(t)}}}return e}const s=i(p),T=P({name:"VitePressApp",setup(){const{site:e,lang:a,dir:t}=R();return w(()=>{y(()=>{document.documentElement.lang=a.value,document.documentElement.dir=t.value})}),e.value.router.prefetchLinks&&C(),b(),E(),s.setup&&s.setup(),()=>S(s.Layout)}});async function D(){globalThis.__VITEPRESS__=!0;const e=j(),a=_();a.provide(c,e);const t=l(e.route);return a.provide(f,t),a.component("Content",d),a.component("ClientOnly",m),Object.defineProperties(a.config.globalProperties,{$frontmatter:{get(){return t.frontmatter.value}},$params:{get(){return t.page.value.params}}}),s.enhanceApp&&await s.enhanceApp({app:a,router:e,siteData:h}),{app:a,router:e,data:t}}function _(){return g(T)}function j(){let e=o,a;return A(t=>{let n=v(t),r=null;return n&&(e&&(a=n),(e||a===n)&&(n=n.replace(/\.js$/,".lean.js")),r=import(n)),o&&(e=!1),r},s.NotFound)}o&&D().then(({app:e,router:a,data:t})=>{a.go().then(()=>{u(a.route,t.site),e.mount("#app")})});export{D as createApp}; +import{R as p}from"./chunks/theme.CG5UfwCm.js";import{R as o,a6 as u,a7 as c,a8 as l,a9 as f,aa as d,ab as m,ac as 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methods"]},"145":{"title":"Utils","titles":[]},"146":{"title":"References","titles":[]},"147":{"title":"Bibliography","titles":[]}},"dirtCount":0,"index":[["μ±nσ",{"2":{"145":1}}],["∑α=1d",{"2":{"123":1,"125":1}}],["∫γiudγ≈|γi|ui",{"2":{"123":1}}],["∈nd",{"2":{"122":1}}],["∏",{"2":{"122":1}}],["κ+1|e^",{"2":{"121":1}}],["κ",{"2":{"121":1,"145":2}}],["κ∈n",{"2":{"121":1}}],["β−12",{"2":{"135":1}}],["β=12",{"2":{"135":1}}],["β=1d",{"2":{"122":1}}],["β",{"2":{"115":2,"116":6,"123":1,"135":2}}],["β∂uα∂xβ∂uβ∂xα",{"2":{"109":1}}],["η=",{"2":{"109":1}}],["η",{"2":{"109":1}}],["ϕ",{"2":{"109":2}}],["τ=luavg",{"2":{"109":1}}],["τ",{"2":{"102":1,"109":1}}],["∇2ψ=∇×ω",{"2":{"108":1}}],["∇2ψ=−ω",{"2":{"108":1}}],["∇⋅",{"2":{"103":1}}],["∇⋅u=0",{"2":{"100":1}}],["∇u⋅n=0",{"2":{"102":1}}],["⋅ndγ+1|o|∫ofdω",{"2":{"101":1}}],["ν3ϵ",{"2":{"109":1}}],["ν",{"2":{"100":1}}],["∂∂xα",{"2":{"122":1}}],["∂o",{"2":{"101":1}}],["∂u∂t+∇⋅",{"2":{"100":1}}],["∂y",{"2":{"97":1}}],["∂x",{"2":{"97":1}}],["∂",{"2":{"97":2}}],["⎢⠈⠻⢷⢗⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⣦⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠁⠱⢅⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⢤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠈⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠐⢑⢔⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠈⠻⣿⣿⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠐⢑⢔⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠈⠛⣟⣽⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠈⠓⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠓⢄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠕⢅⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢽⣦⡀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣗⣄⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠕⢅⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣵⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢑⢔⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢑⢔⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠕⢅⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠻⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠕⢅⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢑⢔⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢑⢔⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠑⢤⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠕⢅⠄⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢑⢆⢀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢵⢷⣦⡀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠕⢅⠄⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣿⣿⣦⡀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣟⣽⣤⡀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⢟⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢽⢦⡀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣕⣄⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠓⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠈⠹⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎡⣄⣤⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎤",{"2":{"97":1}}],["⎡⠀⢄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎤",{"2":{"97":1}}],["⎡⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎤",{"2":{"97":1}}],["⎡⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎤",{"2":{"97":1}}],["⎣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠑⠀⎦",{"2":{"97":1}}],["⎣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠛⠙⎦",{"2":{"97":1}}],["⎣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠙⎦",{"2":{"97":1}}],["⎣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠑⠀⎦",{"2":{"97":1}}],["ψi+h1+3h2−ψi+h1+h2xi1+3",{"2":{"131":1}}],["ψi+3",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methods"]},"145":{"title":"Utils","titles":[]},"146":{"title":"References","titles":[]},"147":{"title":"Bibliography","titles":[]}},"dirtCount":0,"index":[["μ±nσ",{"2":{"145":1}}],["∑α=1d",{"2":{"123":1,"125":1}}],["∫γiudγ≈|γi|ui",{"2":{"123":1}}],["∈nd",{"2":{"122":1}}],["∏",{"2":{"122":1}}],["κ+1|e^",{"2":{"121":1}}],["κ",{"2":{"121":1,"145":2}}],["κ∈n",{"2":{"121":1}}],["β−12",{"2":{"135":1}}],["β=12",{"2":{"135":1}}],["β=1d",{"2":{"122":1}}],["β",{"2":{"115":2,"116":6,"123":1,"135":2}}],["β∂uα∂xβ∂uβ∂xα",{"2":{"109":1}}],["η=",{"2":{"109":1}}],["η",{"2":{"109":1}}],["ϕ",{"2":{"109":2}}],["τ=luavg",{"2":{"109":1}}],["τ",{"2":{"102":1,"109":1}}],["∇2ψ=∇×ω",{"2":{"108":1}}],["∇2ψ=−ω",{"2":{"108":1}}],["∇⋅",{"2":{"103":1}}],["∇⋅u=0",{"2":{"100":1}}],["∇u⋅n=0",{"2":{"102":1}}],["⋅ndγ+1|o|∫ofdω",{"2":{"101":1}}],["ν3ϵ",{"2":{"109":1}}],["ν",{"2":{"100":1}}],["∂∂xα",{"2":{"122":1}}],["∂o",{"2":{"101":1}}],["∂u∂t+∇⋅",{"2":{"100":1}}],["∂y",{"2":{"97":1}}],["∂x",{"2":{"97":1}}],["∂",{"2":{"97":2}}],["⎢⠈⠻⢷⢗⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⣦⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠁⠱⢅⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⢤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠈⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠐⢑⢔⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠈⠻⣿⣿⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠐⢑⢔⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠈⠛⣟⣽⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠈⠓⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠓⢄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠕⢅⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢽⣦⡀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣗⣄⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠕⢅⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣵⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢑⢔⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢑⢔⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠕⢅⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠻⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠕⢅⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢑⢔⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢑⢔⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠑⢤⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠕⢅⠄⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢑⢆⢀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢵⢷⣦⡀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠕⢅⠄⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣿⣿⣦⡀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣟⣽⣤⡀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⢟⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢽⢦⡀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣕⣄⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠀⠓⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎢⠀⠈⠹⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥",{"2":{"97":1}}],["⎡⣄⣤⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎤",{"2":{"97":1}}],["⎡⠀⢄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎤",{"2":{"97":1}}],["⎡⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎤",{"2":{"97":1}}],["⎡⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎤",{"2":{"97":1}}],["⎣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠑⠀⎦",{"2":{"97":1}}],["⎣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠛⠙⎦",{"2":{"97":1}}],["⎣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠙⎦",{"2":{"97":1}}],["⎣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠑⠀⎦",{"2":{"97":1}}],["ψi+h1+3h2−ψi+h1+h2xi1+3",{"2":{"131":1}}],["ψi+3",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k="/IncompressibleNavierStokes.jl/dev/assets/joyoqhm.D0gpz0HH.png",p="/IncompressibleNavierStokes.jl/dev/Actuator2D.mp4",l="/IncompressibleNavierStokes.jl/dev/assets/bbydpmh.B0GX2W7p.png",t="/IncompressibleNavierStokes.jl/dev/assets/vwimpek.BjVPut0E.png",e="/IncompressibleNavierStokes.jl/dev/assets/eimjwfa.BQ4kmQL6.png",C=JSON.parse('{"title":"Unsteady actuator case - 2D","description":"","frontmatter":{},"headers":[],"relativePath":"examples/generated/Actuator2D.md","filePath":"examples/generated/Actuator2D.md","lastUpdated":null}'),E={name:"examples/generated/Actuator2D.md"};function d(r,s,g,y,F,c){return h(),a("div",null,s[0]||(s[0]=[n(`

Unsteady actuator case - 2D ​

In this example, an unsteady inlet velocity profile at encounters a wind turbine blade in a wall-less domain. The blade is modeled as a uniform body force on a thin rectangle.

Packages ​

A Makie plotting backend is needed for plotting. GLMakie creates an interactive window (useful for real-time plotting), but does not work when building this example on GitHub. CairoMakie makes high-quality static vector-graphics plots.

julia
using CairoMakie
+import{_ as i,c as a,a5 as n,o as h}from"./chunks/framework.Dr1kDDE8.js";const k="/IncompressibleNavierStokes.jl/dev/assets/hcmbwbw.D0gpz0HH.png",p="/IncompressibleNavierStokes.jl/dev/Actuator2D.mp4",l="/IncompressibleNavierStokes.jl/dev/assets/giwhfzn.B0GX2W7p.png",t="/IncompressibleNavierStokes.jl/dev/assets/ybniibi.BjVPut0E.png",e="/IncompressibleNavierStokes.jl/dev/assets/zlzjvsb.BQ4kmQL6.png",C=JSON.parse('{"title":"Unsteady actuator case - 2D","description":"","frontmatter":{},"headers":[],"relativePath":"examples/generated/Actuator2D.md","filePath":"examples/generated/Actuator2D.md","lastUpdated":null}'),E={name:"examples/generated/Actuator2D.md"};function d(r,s,g,y,F,c){return h(),a("div",null,s[0]||(s[0]=[n(`
Unsteady actuator case - 2D ​
In this example, an unsteady inlet velocity profile at encounters a wind turbine blade in a wall-less domain. The blade is modeled as a uniform body force on a thin rectangle.
Packages ​
A Makie plotting backend is needed for plotting. GLMakie creates an interactive window (useful for real-time plotting), but does not work when building this example on GitHub. CairoMakie makes high-quality static vector-graphics plots.
julia
using CairoMakie
 using IncompressibleNavierStokes
Setup ​
A 2D grid is a Cartesian product of two vectors
julia
n = 40
 x = LinRange(0.0, 10.0, 5n + 1), LinRange(-2.0, 2.0, 2n + 1)
 plotgrid(x...; figure = (; size = (600, 300)))
Boundary conditions
julia
inflow(dim, x, y, t) = sinpi(sinpi(t / 6) / 6 + (dim == 1) / 2)
@@ -61,16 +61,16 @@ import{_ as i,c as a,a5 as n,o as h}from"./chunks/framework.Dr1kDDE8.js";const k
         rtp = realtimeplotter(; setup, size = (600, 300), nupdate = 5),
         log = timelogger(; nupdate = 24),
     ),
-);
[ Info: t = 1.2	Δt = 0.05	umax = 1.1	itertime = 0.12
-[ Info: t = 2.4	Δt = 0.05	umax = 1	itertime = 0.044
-[ Info: t = 3.6	Δt = 0.05	umax = 1	itertime = 0.009
-[ Info: t = 4.8	Δt = 0.05	umax = 1	itertime = 0.0098
-[ Info: t = 6	Δt = 0.05	umax = 1	itertime = 0.0093
-[ Info: t = 7.2	Δt = 0.05	umax = 1	itertime = 0.0091
-[ Info: t = 8.4	Δt = 0.05	umax = 1	itertime = 0.0096
-[ Info: t = 9.6	Δt = 0.05	umax = 1	itertime = 0.0093
-[ Info: t = 10.8	Δt = 0.05	umax = 1	itertime = 0.0093
-[ Info: t = 12	Δt = 0.05	umax = 1	itertime = 0.0091
Post-process ​
We create a box to visualize the actuator.
julia
box = (
+);
[ Info: t = 1.2	Δt = 0.05	umax = 1.1	itertime = 0.11
+[ Info: t = 2.4	Δt = 0.05	umax = 1	itertime = 0.04
+[ Info: t = 3.6	Δt = 0.05	umax = 1	itertime = 0.0087
+[ Info: t = 4.8	Δt = 0.05	umax = 1	itertime = 0.0094
+[ Info: t = 6	Δt = 0.05	umax = 1	itertime = 0.0087
+[ Info: t = 7.2	Δt = 0.05	umax = 1	itertime = 0.0086
+[ Info: t = 8.4	Δt = 0.05	umax = 1	itertime = 0.0088
+[ Info: t = 9.6	Δt = 0.05	umax = 1	itertime = 0.009
+[ Info: t = 10.8	Δt = 0.05	umax = 1	itertime = 0.0089
+[ Info: t = 12	Δt = 0.05	umax = 1	itertime = 0.0087
Post-process ​
We create a box to visualize the actuator.
julia
box = (
     [xc - δ / 2, xc - δ / 2, xc + δ / 2, xc + δ / 2, xc - δ / 2],
     [yc + D / 2, yc - D / 2, yc - D / 2, yc + D / 2, yc + D / 2],
 )
([1.945, 1.945, 2.055, 2.055, 1.945], [0.5, -0.5, -0.5, 0.5, 0.5])
Plot pressure
julia
fig = fieldplot(state; setup, size = (600, 300), fieldname = :pressure)
diff --git a/dev/assets/examples_generated_Actuator2D.md.CicJvifW.js b/dev/assets/examples_generated_Actuator2D.md.DxDFVKMk.lean.js
similarity index 98%
rename from dev/assets/examples_generated_Actuator2D.md.CicJvifW.js
rename to dev/assets/examples_generated_Actuator2D.md.DxDFVKMk.lean.js
index f6811cbe3..83855edcc 100644
--- a/dev/assets/examples_generated_Actuator2D.md.CicJvifW.js
+++ b/dev/assets/examples_generated_Actuator2D.md.DxDFVKMk.lean.js
@@ -1,4 +1,4 @@
-import{_ as i,c as a,a5 as n,o as h}from"./chunks/framework.Dr1kDDE8.js";const k="/IncompressibleNavierStokes.jl/dev/assets/joyoqhm.D0gpz0HH.png",p="/IncompressibleNavierStokes.jl/dev/Actuator2D.mp4",l="/IncompressibleNavierStokes.jl/dev/assets/bbydpmh.B0GX2W7p.png",t="/IncompressibleNavierStokes.jl/dev/assets/vwimpek.BjVPut0E.png",e="/IncompressibleNavierStokes.jl/dev/assets/eimjwfa.BQ4kmQL6.png",C=JSON.parse('{"title":"Unsteady actuator case - 2D","description":"","frontmatter":{},"headers":[],"relativePath":"examples/generated/Actuator2D.md","filePath":"examples/generated/Actuator2D.md","lastUpdated":null}'),E={name:"examples/generated/Actuator2D.md"};function d(r,s,g,y,F,c){return h(),a("div",null,s[0]||(s[0]=[n(`
Unsteady actuator case - 2D ​
In this example, an unsteady inlet velocity profile at encounters a wind turbine blade in a wall-less domain. The blade is modeled as a uniform body force on a thin rectangle.
Packages ​
A Makie plotting backend is needed for plotting. GLMakie creates an interactive window (useful for real-time plotting), but does not work when building this example on GitHub. CairoMakie makes high-quality static vector-graphics plots.
julia
using CairoMakie
+import{_ as i,c as a,a5 as n,o as h}from"./chunks/framework.Dr1kDDE8.js";const k="/IncompressibleNavierStokes.jl/dev/assets/hcmbwbw.D0gpz0HH.png",p="/IncompressibleNavierStokes.jl/dev/Actuator2D.mp4",l="/IncompressibleNavierStokes.jl/dev/assets/giwhfzn.B0GX2W7p.png",t="/IncompressibleNavierStokes.jl/dev/assets/ybniibi.BjVPut0E.png",e="/IncompressibleNavierStokes.jl/dev/assets/zlzjvsb.BQ4kmQL6.png",C=JSON.parse('{"title":"Unsteady actuator case - 2D","description":"","frontmatter":{},"headers":[],"relativePath":"examples/generated/Actuator2D.md","filePath":"examples/generated/Actuator2D.md","lastUpdated":null}'),E={name:"examples/generated/Actuator2D.md"};function d(r,s,g,y,F,c){return h(),a("div",null,s[0]||(s[0]=[n(`
Unsteady actuator case - 2D ​
In this example, an unsteady inlet velocity profile at encounters a wind turbine blade in a wall-less domain. The blade is modeled as a uniform body force on a thin rectangle.
Packages ​
A Makie plotting backend is needed for plotting. GLMakie creates an interactive window (useful for real-time plotting), but does not work when building this example on GitHub. CairoMakie makes high-quality static vector-graphics plots.
julia
using CairoMakie
 using IncompressibleNavierStokes
Setup ​
A 2D grid is a Cartesian product of two vectors
julia
n = 40
 x = LinRange(0.0, 10.0, 5n + 1), LinRange(-2.0, 2.0, 2n + 1)
 plotgrid(x...; figure = (; size = (600, 300)))
Boundary conditions
julia
inflow(dim, x, y, t) = sinpi(sinpi(t / 6) / 6 + (dim == 1) / 2)
@@ -61,16 +61,16 @@ import{_ as i,c as a,a5 as n,o as h}from"./chunks/framework.Dr1kDDE8.js";const k
         rtp = realtimeplotter(; setup, size = (600, 300), nupdate = 5),
         log = timelogger(; nupdate = 24),
     ),
-);
[ Info: t = 1.2	Δt = 0.05	umax = 1.1	itertime = 0.12
-[ Info: t = 2.4	Δt = 0.05	umax = 1	itertime = 0.044
-[ Info: t = 3.6	Δt = 0.05	umax = 1	itertime = 0.009
-[ Info: t = 4.8	Δt = 0.05	umax = 1	itertime = 0.0098
-[ Info: t = 6	Δt = 0.05	umax = 1	itertime = 0.0093
-[ Info: t = 7.2	Δt = 0.05	umax = 1	itertime = 0.0091
-[ Info: t = 8.4	Δt = 0.05	umax = 1	itertime = 0.0096
-[ Info: t = 9.6	Δt = 0.05	umax = 1	itertime = 0.0093
-[ Info: t = 10.8	Δt = 0.05	umax = 1	itertime = 0.0093
-[ Info: t = 12	Δt = 0.05	umax = 1	itertime = 0.0091
Post-process ​
We create a box to visualize the actuator.
julia
box = (
+);
[ Info: t = 1.2	Δt = 0.05	umax = 1.1	itertime = 0.11
+[ Info: t = 2.4	Δt = 0.05	umax = 1	itertime = 0.04
+[ Info: t = 3.6	Δt = 0.05	umax = 1	itertime = 0.0087
+[ Info: t = 4.8	Δt = 0.05	umax = 1	itertime = 0.0094
+[ Info: t = 6	Δt = 0.05	umax = 1	itertime = 0.0087
+[ Info: t = 7.2	Δt = 0.05	umax = 1	itertime = 0.0086
+[ Info: t = 8.4	Δt = 0.05	umax = 1	itertime = 0.0088
+[ Info: t = 9.6	Δt = 0.05	umax = 1	itertime = 0.009
+[ Info: t = 10.8	Δt = 0.05	umax = 1	itertime = 0.0089
+[ Info: t = 12	Δt = 0.05	umax = 1	itertime = 0.0087
Post-process ​
We create a box to visualize the actuator.
julia
box = (
     [xc - δ / 2, xc - δ / 2, xc + δ / 2, xc + δ / 2, xc - δ / 2],
     [yc + D / 2, yc - D / 2, yc - D / 2, yc + D / 2, yc + D / 2],
 )
([1.945, 1.945, 2.055, 2.055, 1.945], [0.5, -0.5, -0.5, 0.5, 0.5])
Plot pressure
julia
fig = fieldplot(state; setup, size = (600, 300), fieldname = :pressure)
diff --git a/dev/assets/examples_generated_DecayingTurbulence2D.md.BUaxapZ_.js b/dev/assets/examples_generated_DecayingTurbulence2D.md.Di45Ivji.js
similarity index 95%
rename from dev/assets/examples_generated_DecayingTurbulence2D.md.BUaxapZ_.js
rename to dev/assets/examples_generated_DecayingTurbulence2D.md.Di45Ivji.js
index 84690a02f..923f2d889 100644
--- a/dev/assets/examples_generated_DecayingTurbulence2D.md.BUaxapZ_.js
+++ b/dev/assets/examples_generated_DecayingTurbulence2D.md.Di45Ivji.js
@@ -1,4 +1,4 @@
-import{_ as i,c as a,a5 as n,o as p}from"./chunks/framework.Dr1kDDE8.js";const t="/IncompressibleNavierStokes.jl/dev/DecayingTurbulence2D.mp4",l="/IncompressibleNavierStokes.jl/dev/assets/jnffizt.DlMyxgTb.png",e="/IncompressibleNavierStokes.jl/dev/assets/ivcspuy.Pnvm2dOa.png",h="/IncompressibleNavierStokes.jl/dev/assets/pqbunkl.lOgoLSNC.png",u=JSON.parse('{"title":"Decaying Homogeneous Isotropic Turbulence - 2D","description":"","frontmatter":{},"headers":[],"relativePath":"examples/generated/DecayingTurbulence2D.md","filePath":"examples/generated/DecayingTurbulence2D.md","lastUpdated":null}'),k={name:"examples/generated/DecayingTurbulence2D.md"};function E(r,s,d,g,y,o){return p(),a("div",null,s[0]||(s[0]=[n(`
Decaying Homogeneous Isotropic Turbulence - 2D ​
In this example we consider decaying homogeneous isotropic turbulence, similar to the cases considered in [1] and [2]. The initial velocity field is created randomly, but with a specific energy spectrum. Due to viscous dissipation, the turbulent features eventually group to form larger visible eddies.
Packages ​
We just need IncompressibleNavierStokes and a Makie plotting backend.
julia
using CairoMakie
+import{_ as i,c as a,a5 as n,o as p}from"./chunks/framework.Dr1kDDE8.js";const t="/IncompressibleNavierStokes.jl/dev/DecayingTurbulence2D.mp4",l="/IncompressibleNavierStokes.jl/dev/assets/uvwqkqs.LQXaN1Di.png",e="/IncompressibleNavierStokes.jl/dev/assets/dcezsvt.CcecJ8jR.png",h="/IncompressibleNavierStokes.jl/dev/assets/yoqfjhw.OfhdAyXL.png",u=JSON.parse('{"title":"Decaying Homogeneous Isotropic Turbulence - 2D","description":"","frontmatter":{},"headers":[],"relativePath":"examples/generated/DecayingTurbulence2D.md","filePath":"examples/generated/DecayingTurbulence2D.md","lastUpdated":null}'),k={name:"examples/generated/DecayingTurbulence2D.md"};function E(r,s,d,g,y,o){return p(),a("div",null,s[0]||(s[0]=[n(`
Decaying Homogeneous Isotropic Turbulence - 2D ​
In this example we consider decaying homogeneous isotropic turbulence, similar to the cases considered in [1] and [2]. The initial velocity field is created randomly, but with a specific energy spectrum. Due to viscous dissipation, the turbulent features eventually group to form larger visible eddies.
Packages ​
We just need IncompressibleNavierStokes and a Makie plotting backend.
julia
using CairoMakie
 using IncompressibleNavierStokes
Setup
julia
n = 256
 ax = LinRange(0.0, 1.0, n + 1)
 setup = Setup(; x = (ax, ax), Re = 4e3);
@@ -22,13 +22,14 @@ import{_ as i,c as a,a5 as n,o as p}from"./chunks/framework.Dr1kDDE8.js";const t
         ),
         log = timelogger(; nupdate = 100),
     ),
-);
[ Info: t = 0.0997245	Δt = 0.0011	umax = 3.3	itertime = 0.028
-[ Info: t = 0.205518	Δt = 0.0012	umax = 3	itertime = 0.016
-[ Info: t = 0.311364	Δt = 0.001	umax = 3.5	itertime = 0.016
-[ Info: t = 0.431183	Δt = 0.0013	umax = 2.6	itertime = 0.016
-[ Info: t = 0.579297	Δt = 0.0017	umax = 2.1	itertime = 0.016
-[ Info: t = 0.734226	Δt = 0.0014	umax = 2.4	itertime = 0.016
-[ Info: t = 0.873193	Δt = 0.0014	umax = 2.5	itertime = 0.016
Post-process ​
We may visualize or export the computed fields
Energy history
julia
outputs.ehist
Energy spectrum
julia
outputs.espec
Plot field
julia
fieldplot(state; setup)
Copy-pasteable code ​
Below is the full code for this example stripped of comments and output.
julia
using GLMakie
+);
[ Info: t = 0.105436	Δt = 0.0011	umax = 3.2	itertime = 0.026
+[ Info: t = 0.216847	Δt = 0.00098	umax = 3.6	itertime = 0.015
+[ Info: t = 0.339039	Δt = 0.0013	umax = 2.6	itertime = 0.015
+[ Info: t = 0.462407	Δt = 0.0012	umax = 3	itertime = 0.015
+[ Info: t = 0.583124	Δt = 0.0012	umax = 2.9	itertime = 0.015
+[ Info: t = 0.720378	Δt = 0.0014	umax = 2.6	itertime = 0.015
+[ Info: t = 0.846368	Δt = 0.0012	umax = 2.9	itertime = 0.015
+[ Info: t = 0.976317	Δt = 0.0014	umax = 2.5	itertime = 0.015
Post-process ​
We may visualize or export the computed fields
Energy history
julia
outputs.ehist
Energy spectrum
julia
outputs.espec
Plot field
julia
fieldplot(state; setup)
Copy-pasteable code ​
Below is the full code for this example stripped of comments and output.
julia
using GLMakie
 using IncompressibleNavierStokes
 
 n = 256
diff --git a/dev/assets/examples_generated_DecayingTurbulence2D.md.BUaxapZ_.lean.js b/dev/assets/examples_generated_DecayingTurbulence2D.md.Di45Ivji.lean.js
similarity index 95%
rename from dev/assets/examples_generated_DecayingTurbulence2D.md.BUaxapZ_.lean.js
rename to dev/assets/examples_generated_DecayingTurbulence2D.md.Di45Ivji.lean.js
index 84690a02f..923f2d889 100644
--- a/dev/assets/examples_generated_DecayingTurbulence2D.md.BUaxapZ_.lean.js
+++ b/dev/assets/examples_generated_DecayingTurbulence2D.md.Di45Ivji.lean.js
@@ -1,4 +1,4 @@
-import{_ as i,c as a,a5 as n,o as p}from"./chunks/framework.Dr1kDDE8.js";const t="/IncompressibleNavierStokes.jl/dev/DecayingTurbulence2D.mp4",l="/IncompressibleNavierStokes.jl/dev/assets/jnffizt.DlMyxgTb.png",e="/IncompressibleNavierStokes.jl/dev/assets/ivcspuy.Pnvm2dOa.png",h="/IncompressibleNavierStokes.jl/dev/assets/pqbunkl.lOgoLSNC.png",u=JSON.parse('{"title":"Decaying Homogeneous Isotropic Turbulence - 2D","description":"","frontmatter":{},"headers":[],"relativePath":"examples/generated/DecayingTurbulence2D.md","filePath":"examples/generated/DecayingTurbulence2D.md","lastUpdated":null}'),k={name:"examples/generated/DecayingTurbulence2D.md"};function E(r,s,d,g,y,o){return p(),a("div",null,s[0]||(s[0]=[n(`
Decaying Homogeneous Isotropic Turbulence - 2D ​
In this example we consider decaying homogeneous isotropic turbulence, similar to the cases considered in [1] and [2]. The initial velocity field is created randomly, but with a specific energy spectrum. Due to viscous dissipation, the turbulent features eventually group to form larger visible eddies.
Packages ​
We just need IncompressibleNavierStokes and a Makie plotting backend.
julia
using CairoMakie
+import{_ as i,c as a,a5 as n,o as p}from"./chunks/framework.Dr1kDDE8.js";const t="/IncompressibleNavierStokes.jl/dev/DecayingTurbulence2D.mp4",l="/IncompressibleNavierStokes.jl/dev/assets/uvwqkqs.LQXaN1Di.png",e="/IncompressibleNavierStokes.jl/dev/assets/dcezsvt.CcecJ8jR.png",h="/IncompressibleNavierStokes.jl/dev/assets/yoqfjhw.OfhdAyXL.png",u=JSON.parse('{"title":"Decaying Homogeneous Isotropic Turbulence - 2D","description":"","frontmatter":{},"headers":[],"relativePath":"examples/generated/DecayingTurbulence2D.md","filePath":"examples/generated/DecayingTurbulence2D.md","lastUpdated":null}'),k={name:"examples/generated/DecayingTurbulence2D.md"};function E(r,s,d,g,y,o){return p(),a("div",null,s[0]||(s[0]=[n(`
Decaying Homogeneous Isotropic Turbulence - 2D ​
In this example we consider decaying homogeneous isotropic turbulence, similar to the cases considered in [1] and [2]. The initial velocity field is created randomly, but with a specific energy spectrum. Due to viscous dissipation, the turbulent features eventually group to form larger visible eddies.
Packages ​
We just need IncompressibleNavierStokes and a Makie plotting backend.
julia
using CairoMakie
 using IncompressibleNavierStokes
Setup
julia
n = 256
 ax = LinRange(0.0, 1.0, n + 1)
 setup = Setup(; x = (ax, ax), Re = 4e3);
@@ -22,13 +22,14 @@ import{_ as i,c as a,a5 as n,o as p}from"./chunks/framework.Dr1kDDE8.js";const t
         ),
         log = timelogger(; nupdate = 100),
     ),
-);
[ Info: t = 0.0997245	Δt = 0.0011	umax = 3.3	itertime = 0.028
-[ Info: t = 0.205518	Δt = 0.0012	umax = 3	itertime = 0.016
-[ Info: t = 0.311364	Δt = 0.001	umax = 3.5	itertime = 0.016
-[ Info: t = 0.431183	Δt = 0.0013	umax = 2.6	itertime = 0.016
-[ Info: t = 0.579297	Δt = 0.0017	umax = 2.1	itertime = 0.016
-[ Info: t = 0.734226	Δt = 0.0014	umax = 2.4	itertime = 0.016
-[ Info: t = 0.873193	Δt = 0.0014	umax = 2.5	itertime = 0.016
Post-process ​
We may visualize or export the computed fields
Energy history
julia
outputs.ehist
Energy spectrum
julia
outputs.espec
Plot field
julia
fieldplot(state; setup)
Copy-pasteable code ​
Below is the full code for this example stripped of comments and output.
julia
using GLMakie
+);
[ Info: t = 0.105436	Δt = 0.0011	umax = 3.2	itertime = 0.026
+[ Info: t = 0.216847	Δt = 0.00098	umax = 3.6	itertime = 0.015
+[ Info: t = 0.339039	Δt = 0.0013	umax = 2.6	itertime = 0.015
+[ Info: t = 0.462407	Δt = 0.0012	umax = 3	itertime = 0.015
+[ Info: t = 0.583124	Δt = 0.0012	umax = 2.9	itertime = 0.015
+[ Info: t = 0.720378	Δt = 0.0014	umax = 2.6	itertime = 0.015
+[ Info: t = 0.846368	Δt = 0.0012	umax = 2.9	itertime = 0.015
+[ Info: t = 0.976317	Δt = 0.0014	umax = 2.5	itertime = 0.015
Post-process ​
We may visualize or export the computed fields
Energy history
julia
outputs.ehist
Energy spectrum
julia
outputs.espec
Plot field
julia
fieldplot(state; setup)
Copy-pasteable code ​
Below is the full code for this example stripped of comments and output.
julia
using GLMakie
 using IncompressibleNavierStokes
 
 n = 256
diff --git a/dev/assets/examples_generated_RayleighBenard2D.md.Cmmj9k1P.lean.js b/dev/assets/examples_generated_RayleighBenard2D.md.DK8wXqn9.js
similarity index 99%
rename from dev/assets/examples_generated_RayleighBenard2D.md.Cmmj9k1P.lean.js
rename to dev/assets/examples_generated_RayleighBenard2D.md.DK8wXqn9.js
index cf2072d36..612f1039a 100644
--- a/dev/assets/examples_generated_RayleighBenard2D.md.Cmmj9k1P.lean.js
+++ b/dev/assets/examples_generated_RayleighBenard2D.md.DK8wXqn9.js
@@ -1,4 +1,4 @@
-import{_ as i,c as a,a5 as n,o as h}from"./chunks/framework.Dr1kDDE8.js";const t="/IncompressibleNavierStokes.jl/dev/assets/yhyiovy.BxypEGK2.png",l="/IncompressibleNavierStokes.jl/dev/RayleighBenard2D.mp4",k="/IncompressibleNavierStokes.jl/dev/assets/jyvtrrx.C3CW_rGA.png",p="/IncompressibleNavierStokes.jl/dev/assets/giyebju.D8geYuhY.png",C=JSON.parse('{"title":"Rayleigh-Bénard convection (2D)","description":"","frontmatter":{},"headers":[],"relativePath":"examples/generated/RayleighBenard2D.md","filePath":"examples/generated/RayleighBenard2D.md","lastUpdated":null}'),e={name:"examples/generated/RayleighBenard2D.md"};function E(r,s,d,g,y,F){return h(),a("div",null,s[0]||(s[0]=[n(`
Rayleigh-Bénard convection (2D) ​
A hot and a cold plate generate a convection cell in a box.
julia
using CairoMakie
+import{_ as i,c as a,a5 as n,o as h}from"./chunks/framework.Dr1kDDE8.js";const t="/IncompressibleNavierStokes.jl/dev/assets/koitmsv.BxypEGK2.png",l="/IncompressibleNavierStokes.jl/dev/RayleighBenard2D.mp4",k="/IncompressibleNavierStokes.jl/dev/assets/lslbzsj.C3CW_rGA.png",p="/IncompressibleNavierStokes.jl/dev/assets/vyxolzy.D8geYuhY.png",C=JSON.parse('{"title":"Rayleigh-Bénard convection (2D)","description":"","frontmatter":{},"headers":[],"relativePath":"examples/generated/RayleighBenard2D.md","filePath":"examples/generated/RayleighBenard2D.md","lastUpdated":null}'),e={name:"examples/generated/RayleighBenard2D.md"};function E(r,s,d,g,y,F){return h(),a("div",null,s[0]||(s[0]=[n(`
Rayleigh-Bénard convection (2D) ​
A hot and a cold plate generate a convection cell in a box.
julia
using CairoMakie
 using IncompressibleNavierStokes
Output directory for saving results
julia
outdir = joinpath(@__DIR__, "output", "RayleighBenard2D")
"/home/runner/work/IncompressibleNavierStokes.jl/IncompressibleNavierStokes.jl/docs/build/examples/generated/output/RayleighBenard2D"
Hardware
julia
backend = CPU()
 
 # using CUDA, CUDSS
@@ -93,8 +93,8 @@ import{_ as i,c as a,a5 as n,o as h}from"./chunks/framework.Dr1kDDE8.js";const t
     tlims = (T(0), T(20)),
     Δt = T(1e-2),
     processors,
-);
[ Info: t = 10.0001	Δt = 0.01	umax = 0.24	itertime = 0.016
-[ Info: t = 20.0004	Δt = 0.01	umax = 0.55	itertime = 0.013
Nusselt numbers
julia
outputs.nusselt
Average temperature
julia
outputs.avg
Copy-pasteable code ​
Below is the full code for this example stripped of comments and output.
julia
using GLMakie
+);
[ Info: t = 10.0001	Δt = 0.01	umax = 0.24	itertime = 0.015
+[ Info: t = 20.0004	Δt = 0.01	umax = 0.55	itertime = 0.012
Nusselt numbers
julia
outputs.nusselt
Average temperature
julia
outputs.avg
Copy-pasteable code ​
Below is the full code for this example stripped of comments and output.
julia
using GLMakie
 using IncompressibleNavierStokes
 
 outdir = joinpath(@__DIR__, "output", "RayleighBenard2D")
diff --git a/dev/assets/examples_generated_RayleighBenard2D.md.Cmmj9k1P.js b/dev/assets/examples_generated_RayleighBenard2D.md.DK8wXqn9.lean.js
similarity index 99%
rename from dev/assets/examples_generated_RayleighBenard2D.md.Cmmj9k1P.js
rename to dev/assets/examples_generated_RayleighBenard2D.md.DK8wXqn9.lean.js
index cf2072d36..612f1039a 100644
--- a/dev/assets/examples_generated_RayleighBenard2D.md.Cmmj9k1P.js
+++ b/dev/assets/examples_generated_RayleighBenard2D.md.DK8wXqn9.lean.js
@@ -1,4 +1,4 @@
-import{_ as i,c as a,a5 as n,o as h}from"./chunks/framework.Dr1kDDE8.js";const t="/IncompressibleNavierStokes.jl/dev/assets/yhyiovy.BxypEGK2.png",l="/IncompressibleNavierStokes.jl/dev/RayleighBenard2D.mp4",k="/IncompressibleNavierStokes.jl/dev/assets/jyvtrrx.C3CW_rGA.png",p="/IncompressibleNavierStokes.jl/dev/assets/giyebju.D8geYuhY.png",C=JSON.parse('{"title":"Rayleigh-Bénard convection (2D)","description":"","frontmatter":{},"headers":[],"relativePath":"examples/generated/RayleighBenard2D.md","filePath":"examples/generated/RayleighBenard2D.md","lastUpdated":null}'),e={name:"examples/generated/RayleighBenard2D.md"};function E(r,s,d,g,y,F){return h(),a("div",null,s[0]||(s[0]=[n(`
Rayleigh-Bénard convection (2D) ​
A hot and a cold plate generate a convection cell in a box.
julia
using CairoMakie
+import{_ as i,c as a,a5 as n,o as h}from"./chunks/framework.Dr1kDDE8.js";const t="/IncompressibleNavierStokes.jl/dev/assets/koitmsv.BxypEGK2.png",l="/IncompressibleNavierStokes.jl/dev/RayleighBenard2D.mp4",k="/IncompressibleNavierStokes.jl/dev/assets/lslbzsj.C3CW_rGA.png",p="/IncompressibleNavierStokes.jl/dev/assets/vyxolzy.D8geYuhY.png",C=JSON.parse('{"title":"Rayleigh-Bénard convection (2D)","description":"","frontmatter":{},"headers":[],"relativePath":"examples/generated/RayleighBenard2D.md","filePath":"examples/generated/RayleighBenard2D.md","lastUpdated":null}'),e={name:"examples/generated/RayleighBenard2D.md"};function E(r,s,d,g,y,F){return h(),a("div",null,s[0]||(s[0]=[n(`
Rayleigh-Bénard convection (2D) ​
A hot and a cold plate generate a convection cell in a box.
julia
using CairoMakie
 using IncompressibleNavierStokes
Output directory for saving results
julia
outdir = joinpath(@__DIR__, "output", "RayleighBenard2D")
"/home/runner/work/IncompressibleNavierStokes.jl/IncompressibleNavierStokes.jl/docs/build/examples/generated/output/RayleighBenard2D"
Hardware
julia
backend = CPU()
 
 # using CUDA, CUDSS
@@ -93,8 +93,8 @@ import{_ as i,c as a,a5 as n,o as h}from"./chunks/framework.Dr1kDDE8.js";const t
     tlims = (T(0), T(20)),
     Δt = T(1e-2),
     processors,
-);
[ Info: t = 10.0001	Δt = 0.01	umax = 0.24	itertime = 0.016
-[ Info: t = 20.0004	Δt = 0.01	umax = 0.55	itertime = 0.013
Nusselt numbers
julia
outputs.nusselt
Average temperature
julia
outputs.avg
Copy-pasteable code ​
Below is the full code for this example stripped of comments and output.
julia
using GLMakie
+);
[ Info: t = 10.0001	Δt = 0.01	umax = 0.24	itertime = 0.015
+[ Info: t = 20.0004	Δt = 0.01	umax = 0.55	itertime = 0.012
Nusselt numbers
julia
outputs.nusselt
Average temperature
julia
outputs.avg
Copy-pasteable code ​
Below is the full code for this example stripped of comments and output.
julia
using GLMakie
 using IncompressibleNavierStokes
 
 outdir = joinpath(@__DIR__, "output", "RayleighBenard2D")
diff --git a/dev/assets/examples_generated_RayleighTaylor2D.md.C4WmLQbK.js b/dev/assets/examples_generated_RayleighTaylor2D.md.QTiLYjii.js
similarity index 99%
rename from dev/assets/examples_generated_RayleighTaylor2D.md.C4WmLQbK.js
rename to dev/assets/examples_generated_RayleighTaylor2D.md.QTiLYjii.js
index ce9448534..d609a2689 100644
--- a/dev/assets/examples_generated_RayleighTaylor2D.md.C4WmLQbK.js
+++ b/dev/assets/examples_generated_RayleighTaylor2D.md.QTiLYjii.js
@@ -1,4 +1,4 @@
-import{_ as i,c as a,a5 as n,o as h}from"./chunks/framework.Dr1kDDE8.js";const k="/IncompressibleNavierStokes.jl/dev/assets/aeuusgj.C2NI-x6W.png",t="/IncompressibleNavierStokes.jl/dev/RayleighTaylor2D.mp4",F=JSON.parse('{"title":"Rayleigh-Taylor instability in 2D","description":"","frontmatter":{},"headers":[],"relativePath":"examples/generated/RayleighTaylor2D.md","filePath":"examples/generated/RayleighTaylor2D.md","lastUpdated":null}'),p={name:"examples/generated/RayleighTaylor2D.md"};function l(e,s,E,r,d,g){return h(),a("div",null,s[0]||(s[0]=[n(`
Rayleigh-Taylor instability in 2D ​
Two fluids with different temperatures start mixing.
julia
using CairoMakie
+import{_ as i,c as a,a5 as n,o as h}from"./chunks/framework.Dr1kDDE8.js";const k="/IncompressibleNavierStokes.jl/dev/assets/pmidlev.C2NI-x6W.png",t="/IncompressibleNavierStokes.jl/dev/RayleighTaylor2D.mp4",F=JSON.parse('{"title":"Rayleigh-Taylor instability in 2D","description":"","frontmatter":{},"headers":[],"relativePath":"examples/generated/RayleighTaylor2D.md","filePath":"examples/generated/RayleighTaylor2D.md","lastUpdated":null}'),p={name:"examples/generated/RayleighTaylor2D.md"};function l(e,s,E,r,d,g){return h(),a("div",null,s[0]||(s[0]=[n(`
Rayleigh-Taylor instability in 2D ​
Two fluids with different temperatures start mixing.
julia
using CairoMakie
 using IncompressibleNavierStokes
Output directory for saving results
julia
outdir = joinpath(@__DIR__, "output", "RayleighTaylor2D")
"/home/runner/work/IncompressibleNavierStokes.jl/IncompressibleNavierStokes.jl/docs/build/examples/generated/output/RayleighTaylor2D"
Hardware
julia
backend = CPU()
 
 # using CUDA, CUDSS
@@ -35,14 +35,14 @@ import{_ as i,c as a,a5 as n,o as h}from"./chunks/framework.Dr1kDDE8.js";const k
         log = timelogger(; nupdate = 200),
     ),
 );
[ Info: t = 1	Δt = 0.005	umax = 0.1	itertime = 0.011
-[ Info: t = 2	Δt = 0.005	umax = 0.32	itertime = 0.0033
-[ Info: t = 3	Δt = 0.005	umax = 0.78	itertime = 0.0034
-[ Info: t = 4	Δt = 0.005	umax = 0.75	itertime = 0.0033
-[ Info: t = 5	Δt = 0.005	umax = 0.78	itertime = 0.0033
+[ Info: t = 2	Δt = 0.005	umax = 0.32	itertime = 0.0032
+[ Info: t = 3	Δt = 0.005	umax = 0.78	itertime = 0.0032
+[ Info: t = 4	Δt = 0.005	umax = 0.75	itertime = 0.0032
+[ Info: t = 5	Δt = 0.005	umax = 0.78	itertime = 0.0032
 [ Info: t = 6	Δt = 0.005	umax = 0.84	itertime = 0.0033
-[ Info: t = 7	Δt = 0.005	umax = 0.84	itertime = 0.0033
-[ Info: t = 8	Δt = 0.005	umax = 0.89	itertime = 0.0033
-[ Info: t = 9	Δt = 0.005	umax = 1.2	itertime = 0.0033
+[ Info: t = 7	Δt = 0.005	umax = 0.84	itertime = 0.0032
+[ Info: t = 8	Δt = 0.005	umax = 0.89	itertime = 0.0032
+[ Info: t = 9	Δt = 0.005	umax = 1.2	itertime = 0.0032
 [ Info: t = 10	Δt = 0.005	umax = 0.88	itertime = 0.0033
Copy-pasteable code ​
Below is the full code for this example stripped of comments and output.
julia
using GLMakie
 using IncompressibleNavierStokes
 
diff --git a/dev/assets/examples_generated_RayleighTaylor2D.md.C4WmLQbK.lean.js b/dev/assets/examples_generated_RayleighTaylor2D.md.QTiLYjii.lean.js
similarity index 99%
rename from dev/assets/examples_generated_RayleighTaylor2D.md.C4WmLQbK.lean.js
rename to dev/assets/examples_generated_RayleighTaylor2D.md.QTiLYjii.lean.js
index ce9448534..d609a2689 100644
--- a/dev/assets/examples_generated_RayleighTaylor2D.md.C4WmLQbK.lean.js
+++ b/dev/assets/examples_generated_RayleighTaylor2D.md.QTiLYjii.lean.js
@@ -1,4 +1,4 @@
-import{_ as i,c as a,a5 as n,o as h}from"./chunks/framework.Dr1kDDE8.js";const k="/IncompressibleNavierStokes.jl/dev/assets/aeuusgj.C2NI-x6W.png",t="/IncompressibleNavierStokes.jl/dev/RayleighTaylor2D.mp4",F=JSON.parse('{"title":"Rayleigh-Taylor instability in 2D","description":"","frontmatter":{},"headers":[],"relativePath":"examples/generated/RayleighTaylor2D.md","filePath":"examples/generated/RayleighTaylor2D.md","lastUpdated":null}'),p={name:"examples/generated/RayleighTaylor2D.md"};function l(e,s,E,r,d,g){return h(),a("div",null,s[0]||(s[0]=[n(`
Rayleigh-Taylor instability in 2D ​
Two fluids with different temperatures start mixing.
julia
using CairoMakie
+import{_ as i,c as a,a5 as n,o as h}from"./chunks/framework.Dr1kDDE8.js";const k="/IncompressibleNavierStokes.jl/dev/assets/pmidlev.C2NI-x6W.png",t="/IncompressibleNavierStokes.jl/dev/RayleighTaylor2D.mp4",F=JSON.parse('{"title":"Rayleigh-Taylor instability in 2D","description":"","frontmatter":{},"headers":[],"relativePath":"examples/generated/RayleighTaylor2D.md","filePath":"examples/generated/RayleighTaylor2D.md","lastUpdated":null}'),p={name:"examples/generated/RayleighTaylor2D.md"};function l(e,s,E,r,d,g){return h(),a("div",null,s[0]||(s[0]=[n(`
Rayleigh-Taylor instability in 2D ​
Two fluids with different temperatures start mixing.
julia
using CairoMakie
 using IncompressibleNavierStokes
Output directory for saving results
julia
outdir = joinpath(@__DIR__, "output", "RayleighTaylor2D")
"/home/runner/work/IncompressibleNavierStokes.jl/IncompressibleNavierStokes.jl/docs/build/examples/generated/output/RayleighTaylor2D"
Hardware
julia
backend = CPU()
 
 # using CUDA, CUDSS
@@ -35,14 +35,14 @@ import{_ as i,c as a,a5 as n,o as h}from"./chunks/framework.Dr1kDDE8.js";const k
         log = timelogger(; nupdate = 200),
     ),
 );
[ Info: t = 1	Δt = 0.005	umax = 0.1	itertime = 0.011
-[ Info: t = 2	Δt = 0.005	umax = 0.32	itertime = 0.0033
-[ Info: t = 3	Δt = 0.005	umax = 0.78	itertime = 0.0034
-[ Info: t = 4	Δt = 0.005	umax = 0.75	itertime = 0.0033
-[ Info: t = 5	Δt = 0.005	umax = 0.78	itertime = 0.0033
+[ Info: t = 2	Δt = 0.005	umax = 0.32	itertime = 0.0032
+[ Info: t = 3	Δt = 0.005	umax = 0.78	itertime = 0.0032
+[ Info: t = 4	Δt = 0.005	umax = 0.75	itertime = 0.0032
+[ Info: t = 5	Δt = 0.005	umax = 0.78	itertime = 0.0032
 [ Info: t = 6	Δt = 0.005	umax = 0.84	itertime = 0.0033
-[ Info: t = 7	Δt = 0.005	umax = 0.84	itertime = 0.0033
-[ Info: t = 8	Δt = 0.005	umax = 0.89	itertime = 0.0033
-[ Info: t = 9	Δt = 0.005	umax = 1.2	itertime = 0.0033
+[ Info: t = 7	Δt = 0.005	umax = 0.84	itertime = 0.0032
+[ Info: t = 8	Δt = 0.005	umax = 0.89	itertime = 0.0032
+[ Info: t = 9	Δt = 0.005	umax = 1.2	itertime = 0.0032
 [ Info: t = 10	Δt = 0.005	umax = 0.88	itertime = 0.0033
Copy-pasteable code ​
Below is the full code for this example stripped of comments and output.
julia
using GLMakie
 using IncompressibleNavierStokes
 
diff --git a/dev/assets/examples_generated_TaylorGreenVortex2D.md.XLjKfMqJ.js b/dev/assets/examples_generated_TaylorGreenVortex2D.md.BBlb1Y2_.js
similarity index 99%
rename from dev/assets/examples_generated_TaylorGreenVortex2D.md.XLjKfMqJ.js
rename to dev/assets/examples_generated_TaylorGreenVortex2D.md.BBlb1Y2_.js
index d4e9d2fa4..5a4d9cb0a 100644
--- a/dev/assets/examples_generated_TaylorGreenVortex2D.md.XLjKfMqJ.js
+++ b/dev/assets/examples_generated_TaylorGreenVortex2D.md.BBlb1Y2_.js
@@ -1,4 +1,4 @@
-import{_ as l,c as a,j as s,a as h,a5 as t,o as n}from"./chunks/framework.Dr1kDDE8.js";const p="/IncompressibleNavierStokes.jl/dev/assets/izirzso.CGB02zQg.png",m=JSON.parse('{"title":"Convergence study: Taylor-Green vortex (2D)","description":"","frontmatter":{},"headers":[],"relativePath":"examples/generated/TaylorGreenVortex2D.md","filePath":"examples/generated/TaylorGreenVortex2D.md","lastUpdated":null}'),k={name:"examples/generated/TaylorGreenVortex2D.md"},e={class:"MathJax",jax:"SVG",display:"true",style:{direction:"ltr",display:"block","text-align":"center",margin:"1em 0",position:"relative"}},d={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-2.474ex"},xmlns:"http://www.w3.org/2000/svg",width:"33.107ex",height:"6.078ex",role:"img",focusable:"false",viewBox:"0 -1593.3 14633.5 2686.7","aria-hidden":"true"};function r(E,i,T,Q,g,y){return n(),a("div",null,[i[2]||(i[2]=s("h1",{id:"Convergence-study:-Taylor-Green-vortex-(2D)",tabindex:"-1"},[h("Convergence study: Taylor-Green vortex (2D) "),s("a",{class:"header-anchor",href:"#Convergence-study:-Taylor-Green-vortex-(2D)","aria-label":'Permalink to "Convergence study: Taylor-Green vortex (2D) {#Convergence-study:-Taylor-Green-vortex-(2D)}"'},"​")],-1)),i[3]||(i[3]=s("p",null,"In this example we consider the Taylor-Green vortex. In 2D, it has an analytical solution, given by",-1)),s("mjx-container",e,[(n(),a("svg",d,i[0]||(i[0]=[t('',1)]))),i[1]||(i[1]=s("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("mtable",{displaystyle:"true",columnalign:"right left",columnspacing:"0em",rowspacing:"3pt"},[s("mtr",null,[s("mtd",null,[s("msup",null,[s("mi",null,"u"),s("mn",null,"1")]),s("mo",{stretchy:"false"},"("),s("mi",null,"x"),s("mo",null,","),s("mi",null,"y"),s("mo",null,","),s("mi",null,"t"),s("mo",{stretchy:"false"},")")]),s("mtd",null,[s("mi"),s("mo",null,"="),s("mo",null,"−"),s("mi",null,"sin"),s("mo",{"data-mjx-texclass":"NONE"},"⁡"),s("mo",{stretchy:"false"},"("),s("mi",null,"x"),s("mo",{stretchy:"false"},")"),s("mi",null,"cos"),s("mo",{"data-mjx-texclass":"NONE"},"⁡"),s("mo",{stretchy:"false"},"("),s("mi",null,"y"),s("mo",{stretchy:"false"},")"),s("msup",null,[s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",{mathvariant:"normal"},"e")]),s("mrow",{"data-mjx-texclass":"ORD"},[s("mo",null,"−"),s("mn",null,"2"),s("mi",null,"t"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mo",null,"/")]),s("mi",null,"R"),s("mi",null,"e")])])])]),s("mtr",null,[s("mtd",null,[s("msup",null,[s("mi",null,"u"),s("mn",null,"2")]),s("mo",{stretchy:"false"},"("),s("mi",null,"x"),s("mo",null,","),s("mi",null,"y"),s("mo",null,","),s("mi",null,"t"),s("mo",{stretchy:"false"},")")]),s("mtd",null,[s("mi"),s("mo",null,"="),s("mo",null,"+"),s("mi",null,"cos"),s("mo",{"data-mjx-texclass":"NONE"},"⁡"),s("mo",{stretchy:"false"},"("),s("mi",null,"x"),s("mo",{stretchy:"false"},")"),s("mi",null,"sin"),s("mo",{"data-mjx-texclass":"NONE"},"⁡"),s("mo",{stretchy:"false"},"("),s("mi",null,"y"),s("mo",{stretchy:"false"},")"),s("msup",null,[s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",{mathvariant:"normal"},"e")]),s("mrow",{"data-mjx-texclass":"ORD"},[s("mo",null,"−"),s("mn",null,"2"),s("mi",null,"t"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mo",null,"/")]),s("mi",null,"R"),s("mi",null,"e")])])])])])])],-1))]),i[4]||(i[4]=t(`
This allows us to test the convergence of our solver.
julia
using CairoMakie
+import{_ as l,c as a,j as s,a as h,a5 as t,o as n}from"./chunks/framework.Dr1kDDE8.js";const p="/IncompressibleNavierStokes.jl/dev/assets/ebauyfm.CGB02zQg.png",m=JSON.parse('{"title":"Convergence study: Taylor-Green vortex (2D)","description":"","frontmatter":{},"headers":[],"relativePath":"examples/generated/TaylorGreenVortex2D.md","filePath":"examples/generated/TaylorGreenVortex2D.md","lastUpdated":null}'),k={name:"examples/generated/TaylorGreenVortex2D.md"},e={class:"MathJax",jax:"SVG",display:"true",style:{direction:"ltr",display:"block","text-align":"center",margin:"1em 0",position:"relative"}},d={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-2.474ex"},xmlns:"http://www.w3.org/2000/svg",width:"33.107ex",height:"6.078ex",role:"img",focusable:"false",viewBox:"0 -1593.3 14633.5 2686.7","aria-hidden":"true"};function r(E,i,T,Q,g,y){return n(),a("div",null,[i[2]||(i[2]=s("h1",{id:"Convergence-study:-Taylor-Green-vortex-(2D)",tabindex:"-1"},[h("Convergence study: Taylor-Green vortex (2D) "),s("a",{class:"header-anchor",href:"#Convergence-study:-Taylor-Green-vortex-(2D)","aria-label":'Permalink to "Convergence study: Taylor-Green vortex (2D) {#Convergence-study:-Taylor-Green-vortex-(2D)}"'},"​")],-1)),i[3]||(i[3]=s("p",null,"In this example we consider the Taylor-Green vortex. In 2D, it has an analytical solution, given by",-1)),s("mjx-container",e,[(n(),a("svg",d,i[0]||(i[0]=[t('',1)]))),i[1]||(i[1]=s("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("mtable",{displaystyle:"true",columnalign:"right left",columnspacing:"0em",rowspacing:"3pt"},[s("mtr",null,[s("mtd",null,[s("msup",null,[s("mi",null,"u"),s("mn",null,"1")]),s("mo",{stretchy:"false"},"("),s("mi",null,"x"),s("mo",null,","),s("mi",null,"y"),s("mo",null,","),s("mi",null,"t"),s("mo",{stretchy:"false"},")")]),s("mtd",null,[s("mi"),s("mo",null,"="),s("mo",null,"−"),s("mi",null,"sin"),s("mo",{"data-mjx-texclass":"NONE"},"⁡"),s("mo",{stretchy:"false"},"("),s("mi",null,"x"),s("mo",{stretchy:"false"},")"),s("mi",null,"cos"),s("mo",{"data-mjx-texclass":"NONE"},"⁡"),s("mo",{stretchy:"false"},"("),s("mi",null,"y"),s("mo",{stretchy:"false"},")"),s("msup",null,[s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",{mathvariant:"normal"},"e")]),s("mrow",{"data-mjx-texclass":"ORD"},[s("mo",null,"−"),s("mn",null,"2"),s("mi",null,"t"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mo",null,"/")]),s("mi",null,"R"),s("mi",null,"e")])])])]),s("mtr",null,[s("mtd",null,[s("msup",null,[s("mi",null,"u"),s("mn",null,"2")]),s("mo",{stretchy:"false"},"("),s("mi",null,"x"),s("mo",null,","),s("mi",null,"y"),s("mo",null,","),s("mi",null,"t"),s("mo",{stretchy:"false"},")")]),s("mtd",null,[s("mi"),s("mo",null,"="),s("mo",null,"+"),s("mi",null,"cos"),s("mo",{"data-mjx-texclass":"NONE"},"⁡"),s("mo",{stretchy:"false"},"("),s("mi",null,"x"),s("mo",{stretchy:"false"},")"),s("mi",null,"sin"),s("mo",{"data-mjx-texclass":"NONE"},"⁡"),s("mo",{stretchy:"false"},"("),s("mi",null,"y"),s("mo",{stretchy:"false"},")"),s("msup",null,[s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",{mathvariant:"normal"},"e")]),s("mrow",{"data-mjx-texclass":"ORD"},[s("mo",null,"−"),s("mn",null,"2"),s("mi",null,"t"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mo",null,"/")]),s("mi",null,"R"),s("mi",null,"e")])])])])])])],-1))]),i[4]||(i[4]=t(`
This allows us to test the convergence of our solver.
julia
using CairoMakie
 using IncompressibleNavierStokes
 using LinearAlgebra
Output directory
julia
outdir = joinpath(@__DIR__, "output", "TaylorGreenVortex2D")
 ispath(outdir) || mkpath(outdir)
"/home/runner/work/IncompressibleNavierStokes.jl/IncompressibleNavierStokes.jl/docs/build/examples/generated/output/TaylorGreenVortex2D"
Convergence
julia
"""
diff --git a/dev/assets/examples_generated_TaylorGreenVortex2D.md.XLjKfMqJ.lean.js b/dev/assets/examples_generated_TaylorGreenVortex2D.md.BBlb1Y2_.lean.js
similarity index 99%
rename from dev/assets/examples_generated_TaylorGreenVortex2D.md.XLjKfMqJ.lean.js
rename to dev/assets/examples_generated_TaylorGreenVortex2D.md.BBlb1Y2_.lean.js
index d4e9d2fa4..5a4d9cb0a 100644
--- a/dev/assets/examples_generated_TaylorGreenVortex2D.md.XLjKfMqJ.lean.js
+++ b/dev/assets/examples_generated_TaylorGreenVortex2D.md.BBlb1Y2_.lean.js
@@ -1,4 +1,4 @@
-import{_ as l,c as a,j as s,a as h,a5 as t,o as n}from"./chunks/framework.Dr1kDDE8.js";const p="/IncompressibleNavierStokes.jl/dev/assets/izirzso.CGB02zQg.png",m=JSON.parse('{"title":"Convergence study: Taylor-Green vortex (2D)","description":"","frontmatter":{},"headers":[],"relativePath":"examples/generated/TaylorGreenVortex2D.md","filePath":"examples/generated/TaylorGreenVortex2D.md","lastUpdated":null}'),k={name:"examples/generated/TaylorGreenVortex2D.md"},e={class:"MathJax",jax:"SVG",display:"true",style:{direction:"ltr",display:"block","text-align":"center",margin:"1em 0",position:"relative"}},d={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-2.474ex"},xmlns:"http://www.w3.org/2000/svg",width:"33.107ex",height:"6.078ex",role:"img",focusable:"false",viewBox:"0 -1593.3 14633.5 2686.7","aria-hidden":"true"};function r(E,i,T,Q,g,y){return n(),a("div",null,[i[2]||(i[2]=s("h1",{id:"Convergence-study:-Taylor-Green-vortex-(2D)",tabindex:"-1"},[h("Convergence study: Taylor-Green vortex (2D) "),s("a",{class:"header-anchor",href:"#Convergence-study:-Taylor-Green-vortex-(2D)","aria-label":'Permalink to "Convergence study: Taylor-Green vortex (2D) {#Convergence-study:-Taylor-Green-vortex-(2D)}"'},"​")],-1)),i[3]||(i[3]=s("p",null,"In this example we consider the Taylor-Green vortex. In 2D, it has an analytical solution, given by",-1)),s("mjx-container",e,[(n(),a("svg",d,i[0]||(i[0]=[t('',1)]))),i[1]||(i[1]=s("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("mtable",{displaystyle:"true",columnalign:"right left",columnspacing:"0em",rowspacing:"3pt"},[s("mtr",null,[s("mtd",null,[s("msup",null,[s("mi",null,"u"),s("mn",null,"1")]),s("mo",{stretchy:"false"},"("),s("mi",null,"x"),s("mo",null,","),s("mi",null,"y"),s("mo",null,","),s("mi",null,"t"),s("mo",{stretchy:"false"},")")]),s("mtd",null,[s("mi"),s("mo",null,"="),s("mo",null,"−"),s("mi",null,"sin"),s("mo",{"data-mjx-texclass":"NONE"},"⁡"),s("mo",{stretchy:"false"},"("),s("mi",null,"x"),s("mo",{stretchy:"false"},")"),s("mi",null,"cos"),s("mo",{"data-mjx-texclass":"NONE"},"⁡"),s("mo",{stretchy:"false"},"("),s("mi",null,"y"),s("mo",{stretchy:"false"},")"),s("msup",null,[s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",{mathvariant:"normal"},"e")]),s("mrow",{"data-mjx-texclass":"ORD"},[s("mo",null,"−"),s("mn",null,"2"),s("mi",null,"t"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mo",null,"/")]),s("mi",null,"R"),s("mi",null,"e")])])])]),s("mtr",null,[s("mtd",null,[s("msup",null,[s("mi",null,"u"),s("mn",null,"2")]),s("mo",{stretchy:"false"},"("),s("mi",null,"x"),s("mo",null,","),s("mi",null,"y"),s("mo",null,","),s("mi",null,"t"),s("mo",{stretchy:"false"},")")]),s("mtd",null,[s("mi"),s("mo",null,"="),s("mo",null,"+"),s("mi",null,"cos"),s("mo",{"data-mjx-texclass":"NONE"},"⁡"),s("mo",{stretchy:"false"},"("),s("mi",null,"x"),s("mo",{stretchy:"false"},")"),s("mi",null,"sin"),s("mo",{"data-mjx-texclass":"NONE"},"⁡"),s("mo",{stretchy:"false"},"("),s("mi",null,"y"),s("mo",{stretchy:"false"},")"),s("msup",null,[s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",{mathvariant:"normal"},"e")]),s("mrow",{"data-mjx-texclass":"ORD"},[s("mo",null,"−"),s("mn",null,"2"),s("mi",null,"t"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mo",null,"/")]),s("mi",null,"R"),s("mi",null,"e")])])])])])])],-1))]),i[4]||(i[4]=t(`
This allows us to test the convergence of our solver.
julia
using CairoMakie
+import{_ as l,c as a,j as s,a as h,a5 as t,o as n}from"./chunks/framework.Dr1kDDE8.js";const p="/IncompressibleNavierStokes.jl/dev/assets/ebauyfm.CGB02zQg.png",m=JSON.parse('{"title":"Convergence study: Taylor-Green vortex (2D)","description":"","frontmatter":{},"headers":[],"relativePath":"examples/generated/TaylorGreenVortex2D.md","filePath":"examples/generated/TaylorGreenVortex2D.md","lastUpdated":null}'),k={name:"examples/generated/TaylorGreenVortex2D.md"},e={class:"MathJax",jax:"SVG",display:"true",style:{direction:"ltr",display:"block","text-align":"center",margin:"1em 0",position:"relative"}},d={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-2.474ex"},xmlns:"http://www.w3.org/2000/svg",width:"33.107ex",height:"6.078ex",role:"img",focusable:"false",viewBox:"0 -1593.3 14633.5 2686.7","aria-hidden":"true"};function r(E,i,T,Q,g,y){return n(),a("div",null,[i[2]||(i[2]=s("h1",{id:"Convergence-study:-Taylor-Green-vortex-(2D)",tabindex:"-1"},[h("Convergence study: Taylor-Green vortex (2D) "),s("a",{class:"header-anchor",href:"#Convergence-study:-Taylor-Green-vortex-(2D)","aria-label":'Permalink to "Convergence study: Taylor-Green vortex (2D) {#Convergence-study:-Taylor-Green-vortex-(2D)}"'},"​")],-1)),i[3]||(i[3]=s("p",null,"In this example we consider the Taylor-Green vortex. In 2D, it has an analytical solution, given by",-1)),s("mjx-container",e,[(n(),a("svg",d,i[0]||(i[0]=[t('',1)]))),i[1]||(i[1]=s("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("mtable",{displaystyle:"true",columnalign:"right left",columnspacing:"0em",rowspacing:"3pt"},[s("mtr",null,[s("mtd",null,[s("msup",null,[s("mi",null,"u"),s("mn",null,"1")]),s("mo",{stretchy:"false"},"("),s("mi",null,"x"),s("mo",null,","),s("mi",null,"y"),s("mo",null,","),s("mi",null,"t"),s("mo",{stretchy:"false"},")")]),s("mtd",null,[s("mi"),s("mo",null,"="),s("mo",null,"−"),s("mi",null,"sin"),s("mo",{"data-mjx-texclass":"NONE"},"⁡"),s("mo",{stretchy:"false"},"("),s("mi",null,"x"),s("mo",{stretchy:"false"},")"),s("mi",null,"cos"),s("mo",{"data-mjx-texclass":"NONE"},"⁡"),s("mo",{stretchy:"false"},"("),s("mi",null,"y"),s("mo",{stretchy:"false"},")"),s("msup",null,[s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",{mathvariant:"normal"},"e")]),s("mrow",{"data-mjx-texclass":"ORD"},[s("mo",null,"−"),s("mn",null,"2"),s("mi",null,"t"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mo",null,"/")]),s("mi",null,"R"),s("mi",null,"e")])])])]),s("mtr",null,[s("mtd",null,[s("msup",null,[s("mi",null,"u"),s("mn",null,"2")]),s("mo",{stretchy:"false"},"("),s("mi",null,"x"),s("mo",null,","),s("mi",null,"y"),s("mo",null,","),s("mi",null,"t"),s("mo",{stretchy:"false"},")")]),s("mtd",null,[s("mi"),s("mo",null,"="),s("mo",null,"+"),s("mi",null,"cos"),s("mo",{"data-mjx-texclass":"NONE"},"⁡"),s("mo",{stretchy:"false"},"("),s("mi",null,"x"),s("mo",{stretchy:"false"},")"),s("mi",null,"sin"),s("mo",{"data-mjx-texclass":"NONE"},"⁡"),s("mo",{stretchy:"false"},"("),s("mi",null,"y"),s("mo",{stretchy:"false"},")"),s("msup",null,[s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",{mathvariant:"normal"},"e")]),s("mrow",{"data-mjx-texclass":"ORD"},[s("mo",null,"−"),s("mn",null,"2"),s("mi",null,"t"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mo",null,"/")]),s("mi",null,"R"),s("mi",null,"e")])])])])])])],-1))]),i[4]||(i[4]=t(`
This allows us to test the convergence of our solver.
julia
using CairoMakie
 using IncompressibleNavierStokes
 using LinearAlgebra
Output directory
julia
outdir = joinpath(@__DIR__, "output", "TaylorGreenVortex2D")
 ispath(outdir) || mkpath(outdir)
"/home/runner/work/IncompressibleNavierStokes.jl/IncompressibleNavierStokes.jl/docs/build/examples/generated/output/TaylorGreenVortex2D"
Convergence
julia
"""
diff --git a/dev/assets/bbydpmh.B0GX2W7p.png b/dev/assets/giwhfzn.B0GX2W7p.png
similarity index 100%
rename from dev/assets/bbydpmh.B0GX2W7p.png
rename to dev/assets/giwhfzn.B0GX2W7p.png
diff --git a/dev/assets/joyoqhm.D0gpz0HH.png b/dev/assets/hcmbwbw.D0gpz0HH.png
similarity index 100%
rename from dev/assets/joyoqhm.D0gpz0HH.png
rename to dev/assets/hcmbwbw.D0gpz0HH.png
diff --git a/dev/assets/ivcspuy.Pnvm2dOa.png b/dev/assets/ivcspuy.Pnvm2dOa.png
deleted file mode 100644
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Neural closure models ​
NeuralClosure
These features are experimental, and require cloning IncompressibleNavierStokes from GitHub:
sh
git clone https://github.com/agdestein/IncompressibleNavierStokes.jl
-cd IncompressibleNavierStokes/lib/NeuralClosure
Large eddy simulation, a closure model is required. With IncompressibleNavierStokes, a neural closure model can be trained on filtered DNS data. The discrete DNS equations are given by
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julia
collocate(u) -> Any
Interpolate velocity components to volume centers.
source
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julia
create_closure(layers...; rng)
Create neural closure model from layers.
source
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julia
create_tensorclosure(layers...; setup, rng)
Create tensor basis closure.
source
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julia
decollocate(u) -> Any
Interpolate closure force from volume centers to volume faces.
source
',3))]),t("details",q,[t("summary",null,[s[56]||(s[56]=t("a",{id:"NeuralClosure.wrappedclosure-Tuple{Any, Any}",href:"#NeuralClosure.wrappedclosure-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"NeuralClosure.wrappedclosure")],-1)),s[57]||(s[57]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[58]||(s[58]=e(`
julia
wrappedclosure(
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Neural closure models ​
NeuralClosure
These features are experimental, and require cloning IncompressibleNavierStokes from GitHub:
sh
git clone https://github.com/agdestein/IncompressibleNavierStokes.jl
+cd IncompressibleNavierStokes/lib/NeuralClosure
Large eddy simulation, a closure model is required. With IncompressibleNavierStokes, a neural closure model can be trained on filtered DNS data. The discrete DNS equations are given by
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We here assumed that ")),t("mjx-container",j,[(o(),n("svg",C,s[19]||(s[19]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D440",d:"M289 629Q289 635 232 637Q208 637 201 638T194 648Q194 649 196 659Q197 662 198 666T199 671T201 676T203 679T207 681T212 683T220 683T232 684Q238 684 262 684T307 683Q386 683 398 683T414 678Q415 674 451 396L487 117L510 154Q534 190 574 254T662 394Q837 673 839 675Q840 676 842 678T846 681L852 683H948Q965 683 988 683T1017 684Q1051 684 1051 673Q1051 668 1048 656T1045 643Q1041 637 1008 637Q968 636 957 634T939 623Q936 618 867 340T797 59Q797 55 798 54T805 50T822 48T855 46H886Q892 37 892 35Q892 19 885 5Q880 0 869 0Q864 0 828 1T736 2Q675 2 644 2T609 1Q592 1 592 11Q592 13 594 25Q598 41 602 43T625 46Q652 46 685 49Q699 52 704 61Q706 65 742 207T813 490T848 631L654 322Q458 10 453 5Q451 4 449 3Q444 0 433 0Q418 0 415 7Q413 11 374 317L335 624L267 354Q200 88 200 79Q206 46 272 46H282Q288 41 289 37T286 19Q282 3 278 1Q274 0 267 0Q265 0 255 0T221 1T157 2Q127 2 95 1T58 0Q43 0 39 2T35 11Q35 13 38 25T43 40Q45 46 65 46Q135 46 154 86Q158 92 223 354T289 629Z",style:{"stroke-width":"3"}})])])],-1)]))),s[20]||(s[20]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"M")])],-1))]),s[34]||(s[34]=a(" and ")),t("mjx-container",M,[(o(),n("svg",L,s[21]||(s[21]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"3A6",d:"M312 622Q310 623 307 625T303 629T297 631T286 634T270 635T246 636T211 637H184V683H196Q220 680 361 680T526 683H538V637H511Q468 637 447 635T422 631T411 622V533L425 531Q525 519 595 466T665 342Q665 301 642 267T583 209T506 172T425 152L411 150V61Q417 55 421 53T447 48T511 46H538V0H526Q502 3 361 3T196 0H184V46H211Q231 46 245 46T270 47T286 48T297 51T303 54T307 57T312 61V150H310Q309 151 289 153T232 166T160 195Q149 201 136 210T103 238T69 284T56 342Q56 414 128 467T294 530Q309 532 310 533H312V622ZM170 342Q170 207 307 188H312V495H309Q301 495 282 491T231 469T186 423Q170 389 170 342ZM415 188Q487 199 519 236T551 342Q551 384 539 414T507 459T470 481T434 491T415 495H410V188H415Z",style:{"stroke-width":"3"}})])])],-1)]))),s[22]||(s[22]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",{mathvariant:"normal"},"Φ")])],-1))]),s[35]||(s[35]=a(" commute, which is the case for face averaging filters. Replacing ")),t("mjx-container",V,[(o(),n("svg",N,s[23]||(s[23]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D450",d:"M34 159Q34 268 120 355T306 442Q362 442 394 418T427 355Q427 326 408 306T360 285Q341 285 330 295T319 325T330 359T352 380T366 386H367Q367 388 361 392T340 400T306 404Q276 404 249 390Q228 381 206 359Q162 315 142 235T121 119Q121 73 147 50Q169 26 205 26H209Q321 26 394 111Q403 121 406 121Q410 121 419 112T429 98T420 83T391 55T346 25T282 0T202 -11Q127 -11 81 37T34 159Z",style:{"stroke-width":"3"}})])])],-1)]))),s[24]||(s[24]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"c")])],-1))]),s[36]||(s[36]=a(" with a parameterized closure model ")),t("mjx-container",Z,[(o(),n("svg",D,s[25]||(s[25]=[e('',1)]))),s[26]||(s[26]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"m"),t("mo",{stretchy:"false"},"("),t("mrow",{"data-mjx-texclass":"ORD"},[t("mover",null,[t("mi",null,"u"),t("mo",{stretchy:"false"},"¯")])]),t("mo",null,","),t("mi",null,"θ"),t("mo",{stretchy:"false"},")"),t("mo",null,"≈"),t("mi",null,"c")])],-1))]),s[37]||(s[37]=a(" gives the LES equations for the approximate large scale velocity ")),t("mjx-container",F,[(o(),n("svg",A,s[27]||(s[27]=[e('',1)]))),s[28]||(s[28]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mover",null,[t("mi",null,"v"),t("mo",{stretchy:"false"},"¯")])]),t("mo",null,"≈"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mover",null,[t("mi",null,"u"),t("mo",{stretchy:"false"},"¯")])])])],-1))])]),t("mjx-container",O,[(o(),n("svg",S,s[38]||(s[38]=[e('',1)]))),s[39]||(s[39]=t("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[t("mtable",{displaystyle:"true",columnalign:"right 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tools.",-1)),s[43]||(s[43]=t("p",null,[t("a",{href:"https://github.com/agdestein/IncompressibleNavierStokes.jl/blob/a63d20e3a410f14ba6cc612a27e89f8974019d9f/lib/NeuralClosure/src/NeuralClosure.jl#L1-L3",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",R,[t("summary",null,[s[44]||(s[44]=t("a",{id:"NeuralClosure.collocate-Tuple{Any}",href:"#NeuralClosure.collocate-Tuple{Any}"},[t("span",{class:"jlbinding"},"NeuralClosure.collocate")],-1)),s[45]||(s[45]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[46]||(s[46]=e('
julia
collocate(u) -> Any
Interpolate velocity components to volume centers.
source
',3))]),t("details",I,[t("summary",null,[s[47]||(s[47]=t("a",{id:"NeuralClosure.create_closure-Tuple",href:"#NeuralClosure.create_closure-Tuple"},[t("span",{class:"jlbinding"},"NeuralClosure.create_closure")],-1)),s[48]||(s[48]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[49]||(s[49]=e('
julia
create_closure(layers...; rng)
Create neural closure model from layers.
source
',3))]),t("details",z,[t("summary",null,[s[50]||(s[50]=t("a",{id:"NeuralClosure.create_tensorclosure-Tuple",href:"#NeuralClosure.create_tensorclosure-Tuple"},[t("span",{class:"jlbinding"},"NeuralClosure.create_tensorclosure")],-1)),s[51]||(s[51]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[52]||(s[52]=e('
julia
create_tensorclosure(layers...; setup, rng)
Create tensor basis closure.
source
',3))]),t("details",G,[t("summary",null,[s[53]||(s[53]=t("a",{id:"NeuralClosure.decollocate-Tuple{Any}",href:"#NeuralClosure.decollocate-Tuple{Any}"},[t("span",{class:"jlbinding"},"NeuralClosure.decollocate")],-1)),s[54]||(s[54]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[55]||(s[55]=e('
julia
decollocate(u) -> Any
Interpolate closure force from volume centers to volume faces.
source
',3))]),t("details",q,[t("summary",null,[s[56]||(s[56]=t("a",{id:"NeuralClosure.wrappedclosure-Tuple{Any, Any}",href:"#NeuralClosure.wrappedclosure-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"NeuralClosure.wrappedclosure")],-1)),s[57]||(s[57]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[58]||(s[58]=e(`
julia
wrappedclosure(
     m,
     setup
-) -> NeuralClosure.var"#neuralclosure#1"
Wrap closure model and parameters so that it can be used in the solver.
source
`,3))]),s[172]||(s[172]=t("h2",{id:"filters",tabindex:"-1"},[a("Filters "),t("a",{class:"header-anchor",href:"#filters","aria-label":'Permalink to "Filters"'},"​")],-1)),s[173]||(s[173]=t("p",null,"The following filters are available:",-1)),t("details",X,[t("summary",null,[s[59]||(s[59]=t("a",{id:"NeuralClosure.AbstractFilter",href:"#NeuralClosure.AbstractFilter"},[t("span",{class:"jlbinding"},"NeuralClosure.AbstractFilter")],-1)),s[60]||(s[60]=a()),i(l,{type:"info",class:"jlObjectType jlType",text:"Type"})]),s[61]||(s[61]=e('
julia
abstract type AbstractFilter
Discrete DNS filter.
Subtypes ConcreteFilter should implement the in-place method:
(::ConcreteFilter)(v, u, setup_les, compression)
which filters the DNS field u and put result in LES field v. Then the out-of place method:
(::ConcreteFilter)(u, setup_les, compression)
automatically becomes available.
Fields
source
',9))]),t("details",J,[t("summary",null,[s[62]||(s[62]=t("a",{id:"NeuralClosure.FaceAverage",href:"#NeuralClosure.FaceAverage"},[t("span",{class:"jlbinding"},"NeuralClosure.FaceAverage")],-1)),s[63]||(s[63]=a()),i(l,{type:"info",class:"jlObjectType jlType",text:"Type"})]),s[64]||(s[64]=e('
julia
struct FaceAverage <: NeuralClosure.AbstractFilter
Average fine grid velocity field over coarse volume face.
Fields
source
',4))]),t("details",P,[t("summary",null,[s[65]||(s[65]=t("a",{id:"NeuralClosure.VolumeAverage",href:"#NeuralClosure.VolumeAverage"},[t("span",{class:"jlbinding"},"NeuralClosure.VolumeAverage")],-1)),s[66]||(s[66]=a()),i(l,{type:"info",class:"jlObjectType jlType",text:"Type"})]),s[67]||(s[67]=e('
julia
struct VolumeAverage <: NeuralClosure.AbstractFilter
Average fine grid velocity field over coarse volume.
Fields
source
',4))]),t("details",U,[t("summary",null,[s[68]||(s[68]=t("a",{id:"NeuralClosure.reconstruct!-NTuple{5, Any}",href:"#NeuralClosure.reconstruct!-NTuple{5, Any}"},[t("span",{class:"jlbinding"},"NeuralClosure.reconstruct!")],-1)),s[69]||(s[69]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[70]||(s[70]=e('
julia
reconstruct!(u, v, setup_dns, setup_les, comp) -> Any
Reconstruct DNS velocity u from LES velocity v.
source
',3))]),t("details",W,[t("summary",null,[s[71]||(s[71]=t("a",{id:"NeuralClosure.reconstruct-NTuple{4, Any}",href:"#NeuralClosure.reconstruct-NTuple{4, Any}"},[t("span",{class:"jlbinding"},"NeuralClosure.reconstruct")],-1)),s[72]||(s[72]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[73]||(s[73]=e('
julia
reconstruct(v, setup_dns, setup_les, comp) -> Any
Reconstruct DNS velocity field. See also reconstruct!.
source
',3))]),s[174]||(s[174]=t("h2",{id:"training",tabindex:"-1"},[a("Training "),t("a",{class:"header-anchor",href:"#training","aria-label":'Permalink to "Training"'},"​")],-1)),t("p",null,[s[78]||(s[78]=a("To improve the model parameters, we exploit exact filtered DNS data ")),t("mjx-container",$,[(o(),n("svg",K,s[74]||(s[74]=[e('',1)]))),s[75]||(s[75]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mover",null,[t("mi",null,"u"),t("mo",{stretchy:"false"},"¯")])])])],-1))]),s[79]||(s[79]=a(" and exact commutator errors ")),t("mjx-container",Y,[(o(),n("svg",_,s[76]||(s[76]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D450",d:"M34 159Q34 268 120 355T306 442Q362 442 394 418T427 355Q427 326 408 306T360 285Q341 285 330 295T319 325T330 359T352 380T366 386H367Q367 388 361 392T340 400T306 404Q276 404 249 390Q228 381 206 359Q162 315 142 235T121 119Q121 73 147 50Q169 26 205 26H209Q321 26 394 111Q403 121 406 121Q410 121 419 112T429 98T420 83T391 55T346 25T282 0T202 -11Q127 -11 81 37T34 159Z",style:{"stroke-width":"3"}})])])],-1)]))),s[77]||(s[77]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"c")])],-1))]),s[80]||(s[80]=a(" obtained through DNS. The model is trained by minimizing the a priori loss function"))]),t("mjx-container",t1,[(o(),n("svg",s1,s[81]||(s[81]=[e('',1)]))),s[82]||(s[82]=t("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[t("msup",null,[t("mi",null,"L"),t("mtext",null,"prior")]),t("mo",{stretchy:"false"},"("),t("mi",null,"θ"),t("mo",{stretchy:"false"},")"),t("mo",null,"="),t("mo",{"data-mjx-texclass":"ORD"},"∥"),t("mi",null,"m"),t("mo",{stretchy:"false"},"("),t("mrow",{"data-mjx-texclass":"ORD"},[t("mover",null,[t("mi",null,"u"),t("mo",{stretchy:"false"},"¯")])]),t("mo",null,","),t("mi",null,"θ"),t("mo",{stretchy:"false"},")"),t("mo",null,"−"),t("mi",null,"c"),t("msup",null,[t("mo",{"data-mjx-texclass":"ORD"},"∥"),t("mn",null,"2")]),t("mo",null,",")])],-1))]),s[175]||(s[175]=t("p",null,"or the a posteriori loss function",-1)),t("mjx-container",a1,[(o(),n("svg",e1,s[83]||(s[83]=[e('',1)]))),s[84]||(s[84]=t("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[t("msup",null,[t("mi",null,"L"),t("mtext",null,"post")]),t("mo",{stretchy:"false"},"("),t("mi",null,"θ"),t("mo",{stretchy:"false"},")"),t("mo",null,"="),t("mo",{"data-mjx-texclass":"ORD"},"∥"),t("msub",null,[t("mrow",{"data-mjx-texclass":"ORD"},[t("mover",null,[t("mi",null,"v"),t("mo",{stretchy:"false"},"¯")])]),t("mi",null,"θ")]),t("mo",null,"−"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mover",null,[t("mi",null,"u"),t("mo",{stretchy:"false"},"¯")])]),t("msup",null,[t("mo",{"data-mjx-texclass":"ORD"},"∥"),t("mn",null,"2")]),t("mo",null,",")])],-1))]),t("p",null,[s[91]||(s[91]=a("where ")),t("mjx-container",l1,[(o(),n("svg",i1,s[85]||(s[85]=[e('',1)]))),s[86]||(s[86]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mrow",{"data-mjx-texclass":"ORD"},[t("mover",null,[t("mi",null,"v"),t("mo",{stretchy:"false"},"¯")])]),t("mi",null,"θ")])])],-1))]),s[92]||(s[92]=a(" is the solution to the LES equation for the given parameters ")),t("mjx-container",n1,[(o(),n("svg",o1,s[87]||(s[87]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D703",d:"M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z",style:{"stroke-width":"3"}})])])],-1)]))),s[88]||(s[88]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"θ")])],-1))]),s[93]||(s[93]=a(". The prior loss is easy to evaluate and easy to differentiate, as it does not involve solving the ODE. However, minimizing ")),t("mjx-container",r1,[(o(),n("svg",d1,s[89]||(s[89]=[e('',1)]))),s[90]||(s[90]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msup",null,[t("mi",null,"L"),t("mtext",null,"prior")])])],-1))]),s[94]||(s[94]=a(" does not take into account the effect of the prediction error on the LES solution error. The posterior loss does, but has a longer computational chain involving solving the LES ODE."))]),t("details",p1,[t("summary",null,[s[95]||(s[95]=t("a",{id:"NeuralClosure.create_callback-Tuple{Any}",href:"#NeuralClosure.create_callback-Tuple{Any}"},[t("span",{class:"jlbinding"},"NeuralClosure.create_callback")],-1)),s[96]||(s[96]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[97]||(s[97]=e(`
julia
create_callback(
+) -> NeuralClosure.var"#neuralclosure#1"
Wrap closure model and parameters so that it can be used in the solver.
source
`,3))]),s[172]||(s[172]=t("h2",{id:"filters",tabindex:"-1"},[a("Filters "),t("a",{class:"header-anchor",href:"#filters","aria-label":'Permalink to "Filters"'},"​")],-1)),s[173]||(s[173]=t("p",null,"The following filters are available:",-1)),t("details",X,[t("summary",null,[s[59]||(s[59]=t("a",{id:"NeuralClosure.AbstractFilter",href:"#NeuralClosure.AbstractFilter"},[t("span",{class:"jlbinding"},"NeuralClosure.AbstractFilter")],-1)),s[60]||(s[60]=a()),i(l,{type:"info",class:"jlObjectType jlType",text:"Type"})]),s[61]||(s[61]=e('
julia
abstract type AbstractFilter
Discrete DNS filter.
Subtypes ConcreteFilter should implement the in-place method:
(::ConcreteFilter)(v, u, setup_les, compression)
which filters the DNS field u and put result in LES field v. Then the out-of place method:
(::ConcreteFilter)(u, setup_les, compression)
automatically becomes available.
Fields
source
',9))]),t("details",J,[t("summary",null,[s[62]||(s[62]=t("a",{id:"NeuralClosure.FaceAverage",href:"#NeuralClosure.FaceAverage"},[t("span",{class:"jlbinding"},"NeuralClosure.FaceAverage")],-1)),s[63]||(s[63]=a()),i(l,{type:"info",class:"jlObjectType jlType",text:"Type"})]),s[64]||(s[64]=e('
julia
struct FaceAverage <: NeuralClosure.AbstractFilter
Average fine grid velocity field over coarse volume face.
Fields
source
',4))]),t("details",P,[t("summary",null,[s[65]||(s[65]=t("a",{id:"NeuralClosure.VolumeAverage",href:"#NeuralClosure.VolumeAverage"},[t("span",{class:"jlbinding"},"NeuralClosure.VolumeAverage")],-1)),s[66]||(s[66]=a()),i(l,{type:"info",class:"jlObjectType jlType",text:"Type"})]),s[67]||(s[67]=e('
julia
struct VolumeAverage <: NeuralClosure.AbstractFilter
Average fine grid velocity field over coarse volume.
Fields
source
',4))]),t("details",U,[t("summary",null,[s[68]||(s[68]=t("a",{id:"NeuralClosure.reconstruct!-NTuple{5, Any}",href:"#NeuralClosure.reconstruct!-NTuple{5, Any}"},[t("span",{class:"jlbinding"},"NeuralClosure.reconstruct!")],-1)),s[69]||(s[69]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[70]||(s[70]=e('
julia
reconstruct!(u, v, setup_dns, setup_les, comp) -> Any
Reconstruct DNS velocity u from LES velocity v.
source
',3))]),t("details",W,[t("summary",null,[s[71]||(s[71]=t("a",{id:"NeuralClosure.reconstruct-NTuple{4, Any}",href:"#NeuralClosure.reconstruct-NTuple{4, Any}"},[t("span",{class:"jlbinding"},"NeuralClosure.reconstruct")],-1)),s[72]||(s[72]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[73]||(s[73]=e('
julia
reconstruct(v, setup_dns, setup_les, comp) -> Any
Reconstruct DNS velocity field. See also reconstruct!.
source
',3))]),s[174]||(s[174]=t("h2",{id:"training",tabindex:"-1"},[a("Training "),t("a",{class:"header-anchor",href:"#training","aria-label":'Permalink to "Training"'},"​")],-1)),t("p",null,[s[78]||(s[78]=a("To improve the model parameters, we exploit exact filtered DNS data ")),t("mjx-container",$,[(o(),n("svg",K,s[74]||(s[74]=[e('',1)]))),s[75]||(s[75]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mover",null,[t("mi",null,"u"),t("mo",{stretchy:"false"},"¯")])])])],-1))]),s[79]||(s[79]=a(" and exact commutator errors ")),t("mjx-container",Y,[(o(),n("svg",_,s[76]||(s[76]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D450",d:"M34 159Q34 268 120 355T306 442Q362 442 394 418T427 355Q427 326 408 306T360 285Q341 285 330 295T319 325T330 359T352 380T366 386H367Q367 388 361 392T340 400T306 404Q276 404 249 390Q228 381 206 359Q162 315 142 235T121 119Q121 73 147 50Q169 26 205 26H209Q321 26 394 111Q403 121 406 121Q410 121 419 112T429 98T420 83T391 55T346 25T282 0T202 -11Q127 -11 81 37T34 159Z",style:{"stroke-width":"3"}})])])],-1)]))),s[77]||(s[77]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"c")])],-1))]),s[80]||(s[80]=a(" obtained through DNS. The model is trained by minimizing the a priori loss function"))]),t("mjx-container",t1,[(o(),n("svg",s1,s[81]||(s[81]=[e('',1)]))),s[82]||(s[82]=t("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[t("msup",null,[t("mi",null,"L"),t("mtext",null,"prior")]),t("mo",{stretchy:"false"},"("),t("mi",null,"θ"),t("mo",{stretchy:"false"},")"),t("mo",null,"="),t("mo",{"data-mjx-texclass":"ORD"},"∥"),t("mi",null,"m"),t("mo",{stretchy:"false"},"("),t("mrow",{"data-mjx-texclass":"ORD"},[t("mover",null,[t("mi",null,"u"),t("mo",{stretchy:"false"},"¯")])]),t("mo",null,","),t("mi",null,"θ"),t("mo",{stretchy:"false"},")"),t("mo",null,"−"),t("mi",null,"c"),t("msup",null,[t("mo",{"data-mjx-texclass":"ORD"},"∥"),t("mn",null,"2")]),t("mo",null,",")])],-1))]),s[175]||(s[175]=t("p",null,"or the a posteriori loss function",-1)),t("mjx-container",a1,[(o(),n("svg",e1,s[83]||(s[83]=[e('',1)]))),s[84]||(s[84]=t("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[t("msup",null,[t("mi",null,"L"),t("mtext",null,"post")]),t("mo",{stretchy:"false"},"("),t("mi",null,"θ"),t("mo",{stretchy:"false"},")"),t("mo",null,"="),t("mo",{"data-mjx-texclass":"ORD"},"∥"),t("msub",null,[t("mrow",{"data-mjx-texclass":"ORD"},[t("mover",null,[t("mi",null,"v"),t("mo",{stretchy:"false"},"¯")])]),t("mi",null,"θ")]),t("mo",null,"−"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mover",null,[t("mi",null,"u"),t("mo",{stretchy:"false"},"¯")])]),t("msup",null,[t("mo",{"data-mjx-texclass":"ORD"},"∥"),t("mn",null,"2")]),t("mo",null,",")])],-1))]),t("p",null,[s[91]||(s[91]=a("where ")),t("mjx-container",l1,[(o(),n("svg",i1,s[85]||(s[85]=[e('',1)]))),s[86]||(s[86]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mrow",{"data-mjx-texclass":"ORD"},[t("mover",null,[t("mi",null,"v"),t("mo",{stretchy:"false"},"¯")])]),t("mi",null,"θ")])])],-1))]),s[92]||(s[92]=a(" is the solution to the LES equation for the given parameters ")),t("mjx-container",n1,[(o(),n("svg",o1,s[87]||(s[87]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D703",d:"M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z",style:{"stroke-width":"3"}})])])],-1)]))),s[88]||(s[88]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"θ")])],-1))]),s[93]||(s[93]=a(". The prior loss is easy to evaluate and easy to differentiate, as it does not involve solving the ODE. However, minimizing ")),t("mjx-container",r1,[(o(),n("svg",d1,s[89]||(s[89]=[e('',1)]))),s[90]||(s[90]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msup",null,[t("mi",null,"L"),t("mtext",null,"prior")])])],-1))]),s[94]||(s[94]=a(" does not take into account the effect of the prediction error on the LES solution error. The posterior loss does, but has a longer computational chain involving solving the LES ODE."))]),t("details",p1,[t("summary",null,[s[95]||(s[95]=t("a",{id:"NeuralClosure.create_callback-Tuple{Any}",href:"#NeuralClosure.create_callback-Tuple{Any}"},[t("span",{class:"jlbinding"},"NeuralClosure.create_callback")],-1)),s[96]||(s[96]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[97]||(s[97]=e(`
julia
create_callback(
     err;
     θ,
     callbackstate,
@@ -11,29 +11,29 @@ import{_ as r,c as n,a5 as e,j as t,a,G as i,B as d,o}from"./chunks/framework.Dr
     displayupdates,
     figfile,
     nupdate
-)
Create convergence plot for relative error between f(x, θ) and y. At each callback, plot is updated and current error is printed.
If state is nonempty, it also plots previous convergence.
If not using interactive GLMakie window, set displayupdates to true.
source
`,5))]),t("details",T1,[t("summary",null,[s[98]||(s[98]=t("a",{id:"NeuralClosure.create_dataloader_post-Tuple{Any}",href:"#NeuralClosure.create_dataloader_post-Tuple{Any}"},[t("span",{class:"jlbinding"},"NeuralClosure.create_dataloader_post")],-1)),s[99]||(s[99]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[100]||(s[100]=e(`
julia
create_dataloader_post(
+)
Create convergence plot for relative error between f(x, θ) and y. At each callback, plot is updated and current error is printed.
If state is nonempty, it also plots previous convergence.
If not using interactive GLMakie window, set displayupdates to true.
source
`,5))]),t("details",T1,[t("summary",null,[s[98]||(s[98]=t("a",{id:"NeuralClosure.create_dataloader_post-Tuple{Any}",href:"#NeuralClosure.create_dataloader_post-Tuple{Any}"},[t("span",{class:"jlbinding"},"NeuralClosure.create_dataloader_post")],-1)),s[99]||(s[99]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[100]||(s[100]=e(`
julia
create_dataloader_post(
     trajectories;
     ntrajectory,
     nunroll,
     device
-)
Create trajectory dataloader.
source
`,3))]),t("details",Q1,[t("summary",null,[s[101]||(s[101]=t("a",{id:"NeuralClosure.create_dataloader_prior-Tuple{Any}",href:"#NeuralClosure.create_dataloader_prior-Tuple{Any}"},[t("span",{class:"jlbinding"},"NeuralClosure.create_dataloader_prior")],-1)),s[102]||(s[102]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[103]||(s[103]=e(`
julia
create_dataloader_prior(
+)
Create trajectory dataloader.
source
`,3))]),t("details",Q1,[t("summary",null,[s[101]||(s[101]=t("a",{id:"NeuralClosure.create_dataloader_prior-Tuple{Any}",href:"#NeuralClosure.create_dataloader_prior-Tuple{Any}"},[t("span",{class:"jlbinding"},"NeuralClosure.create_dataloader_prior")],-1)),s[102]||(s[102]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[103]||(s[103]=e(`
julia
create_dataloader_prior(
     data;
     batchsize,
     device
-) -> NeuralClosure.var"#dataloader#44"{Int64}
Create dataloader that uses a batch of batchsize random samples from data at each evaluation. The batch is moved to device.
source
`,3))]),t("details",m1,[t("summary",null,[s[104]||(s[104]=t("a",{id:"NeuralClosure.create_loss_post-Tuple{}",href:"#NeuralClosure.create_loss_post-Tuple{}"},[t("span",{class:"jlbinding"},"NeuralClosure.create_loss_post")],-1)),s[105]||(s[105]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[106]||(s[106]=e(`
julia
create_loss_post(
+) -> NeuralClosure.var"#dataloader#44"{Int64}
Create dataloader that uses a batch of batchsize random samples from data at each evaluation. The batch is moved to device.
source
`,3))]),t("details",m1,[t("summary",null,[s[104]||(s[104]=t("a",{id:"NeuralClosure.create_loss_post-Tuple{}",href:"#NeuralClosure.create_loss_post-Tuple{}"},[t("span",{class:"jlbinding"},"NeuralClosure.create_loss_post")],-1)),s[105]||(s[105]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[106]||(s[106]=e(`
julia
create_loss_post(
 ;
     setup,
     method,
     psolver,
     closure,
     nsubstep
-)
Create a-posteriori loss function.
source
`,3))]),t("details",h1,[t("summary",null,[s[107]||(s[107]=t("a",{id:"NeuralClosure.create_loss_prior",href:"#NeuralClosure.create_loss_prior"},[t("span",{class:"jlbinding"},"NeuralClosure.create_loss_prior")],-1)),s[108]||(s[108]=a()),i(l,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[109]||(s[109]=e(`
julia
create_loss_prior(
+)
Create a-posteriori loss function.
source
`,3))]),t("details",h1,[t("summary",null,[s[107]||(s[107]=t("a",{id:"NeuralClosure.create_loss_prior",href:"#NeuralClosure.create_loss_prior"},[t("span",{class:"jlbinding"},"NeuralClosure.create_loss_prior")],-1)),s[108]||(s[108]=a()),i(l,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[109]||(s[109]=e(`
julia
create_loss_prior(
     f
 ) -> NeuralClosure.var"#loss_prior#57"{_A, NeuralClosure.var"#55#56"} where _A
 create_loss_prior(
     f,
     normalize
-) -> NeuralClosure.var"#loss_prior#57"
Return mean squared error loss for the predictor f.
source
`,3))]),t("details",u1,[t("summary",null,[s[110]||(s[110]=t("a",{id:"NeuralClosure.create_relerr_post-Tuple{}",href:"#NeuralClosure.create_relerr_post-Tuple{}"},[t("span",{class:"jlbinding"},"NeuralClosure.create_relerr_post")],-1)),s[111]||(s[111]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[112]||(s[112]=e(`
julia
create_relerr_post(
+) -> NeuralClosure.var"#loss_prior#57"
Return mean squared error loss for the predictor f.
source
`,3))]),t("details",u1,[t("summary",null,[s[110]||(s[110]=t("a",{id:"NeuralClosure.create_relerr_post-Tuple{}",href:"#NeuralClosure.create_relerr_post-Tuple{}"},[t("span",{class:"jlbinding"},"NeuralClosure.create_relerr_post")],-1)),s[111]||(s[111]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[112]||(s[112]=e(`
julia
create_relerr_post(
 ;
     data,
     setup,
@@ -41,7 +41,7 @@ import{_ as r,c as n,a5 as e,j as t,a,G as i,B as d,o}from"./chunks/framework.Dr
     psolver,
     closure_model,
     nsubstep
-)
Create a-posteriori relative error.
source
`,3))]),t("details",c1,[t("summary",null,[s[113]||(s[113]=t("a",{id:"NeuralClosure.create_relerr_prior-Tuple{Any, Any, Any}",href:"#NeuralClosure.create_relerr_prior-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"NeuralClosure.create_relerr_prior")],-1)),s[114]||(s[114]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[115]||(s[115]=e('
julia
create_relerr_prior(f, x, y) -> NeuralClosure.var"#58#59"
Create a-priori error.
source
',3))]),t("details",g1,[t("summary",null,[s[116]||(s[116]=t("a",{id:"NeuralClosure.create_relerr_symmetry_post-Tuple{}",href:"#NeuralClosure.create_relerr_symmetry_post-Tuple{}"},[t("span",{class:"jlbinding"},"NeuralClosure.create_relerr_symmetry_post")],-1)),s[117]||(s[117]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[118]||(s[118]=e(`
julia
create_relerr_symmetry_post(
+)
Create a-posteriori relative error.
source
`,3))]),t("details",g1,[t("summary",null,[s[113]||(s[113]=t("a",{id:"NeuralClosure.create_relerr_prior-Tuple{Any, Any, Any}",href:"#NeuralClosure.create_relerr_prior-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"NeuralClosure.create_relerr_prior")],-1)),s[114]||(s[114]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[115]||(s[115]=e('
julia
create_relerr_prior(f, x, y) -> NeuralClosure.var"#58#59"
Create a-priori error.
source
',3))]),t("details",c1,[t("summary",null,[s[116]||(s[116]=t("a",{id:"NeuralClosure.create_relerr_symmetry_post-Tuple{}",href:"#NeuralClosure.create_relerr_symmetry_post-Tuple{}"},[t("span",{class:"jlbinding"},"NeuralClosure.create_relerr_symmetry_post")],-1)),s[117]||(s[117]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[118]||(s[118]=e(`
julia
create_relerr_symmetry_post(
 ;
     u,
     setup,
@@ -50,7 +50,7 @@ import{_ as r,c as n,a5 as e,j as t,a,G as i,B as d,o}from"./chunks/framework.Dr
     Δt,
     nstep,
     g
-)
Create a-posteriori symmetry error.
source
`,3))]),t("details",k1,[t("summary",null,[s[119]||(s[119]=t("a",{id:"NeuralClosure.create_relerr_symmetry_prior-Tuple{}",href:"#NeuralClosure.create_relerr_symmetry_prior-Tuple{}"},[t("span",{class:"jlbinding"},"NeuralClosure.create_relerr_symmetry_prior")],-1)),s[120]||(s[120]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[121]||(s[121]=e('
julia
create_relerr_symmetry_prior(; u, setup, g)
Create a-priori equivariance error.
source
',3))]),t("details",y1,[t("summary",null,[s[122]||(s[122]=t("a",{id:"NeuralClosure.train-Tuple{}",href:"#NeuralClosure.train-Tuple{}"},[t("span",{class:"jlbinding"},"NeuralClosure.train")],-1)),s[123]||(s[123]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[124]||(s[124]=e(`
julia
train(
+)
Create a-posteriori symmetry error.
source
`,3))]),t("details",k1,[t("summary",null,[s[119]||(s[119]=t("a",{id:"NeuralClosure.create_relerr_symmetry_prior-Tuple{}",href:"#NeuralClosure.create_relerr_symmetry_prior-Tuple{}"},[t("span",{class:"jlbinding"},"NeuralClosure.create_relerr_symmetry_prior")],-1)),s[120]||(s[120]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[121]||(s[121]=e('
julia
create_relerr_symmetry_prior(; u, setup, g)
Create a-priori equivariance error.
source
',3))]),t("details",y1,[t("summary",null,[s[122]||(s[122]=t("a",{id:"NeuralClosure.train-Tuple{}",href:"#NeuralClosure.train-Tuple{}"},[t("span",{class:"jlbinding"},"NeuralClosure.train")],-1)),s[123]||(s[123]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[124]||(s[124]=e(`
julia
train(
 ;
     dataloader,
     loss,
@@ -59,7 +59,7 @@ import{_ as r,c as n,a5 as e,j as t,a,G as i,B as d,o}from"./chunks/framework.Dr
     callback,
     callbackstate,
     λ
-)
Update parameters θ to minimize loss(dataloader(), θ) using the optimiser opt for niter iterations.
Return the a new named tuple (; opt, θ, callbackstate) with updated state and parameters.
source
`,4))]),t("details",b1,[t("summary",null,[s[125]||(s[125]=t("a",{id:"NeuralClosure.trainepoch-Tuple{}",href:"#NeuralClosure.trainepoch-Tuple{}"},[t("span",{class:"jlbinding"},"NeuralClosure.trainepoch")],-1)),s[126]||(s[126]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[127]||(s[127]=e(`
julia
trainepoch(
+)
Update parameters θ to minimize loss(dataloader(), θ) using the optimiser opt for niter iterations.
Return the a new named tuple (; opt, θ, callbackstate) with updated state and parameters.
source
`,4))]),t("details",b1,[t("summary",null,[s[125]||(s[125]=t("a",{id:"NeuralClosure.trainepoch-Tuple{}",href:"#NeuralClosure.trainepoch-Tuple{}"},[t("span",{class:"jlbinding"},"NeuralClosure.trainepoch")],-1)),s[126]||(s[126]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[127]||(s[127]=e(`
julia
trainepoch(
 ;
     dataloader,
     loss,
@@ -69,7 +69,7 @@ import{_ as r,c as n,a5 as e,j as t,a,G as i,B as d,o}from"./chunks/framework.Dr
     device,
     noiselevel,
     λ
-)
Update parameters θ to minimize loss(dataloader(), θ) using the optimiser opt for niter iterations.
Return the a new named tuple (; opt, θ, callbackstate) with updated state and parameters.
source
`,4))]),s[176]||(s[176]=t("h2",{id:"Neural-architectures",tabindex:"-1"},[a("Neural architectures "),t("a",{class:"header-anchor",href:"#Neural-architectures","aria-label":'Permalink to "Neural architectures {#Neural-architectures}"'},"​")],-1)),s[177]||(s[177]=t("p",null,"We provide neural architectures: A convolutional neural network (CNN), group convolutional neural networks (G-CNN) and a Fourier neural operator (FNO).",-1)),t("details",x1,[t("summary",null,[s[128]||(s[128]=t("a",{id:"NeuralClosure.cnn-Tuple{}",href:"#NeuralClosure.cnn-Tuple{}"},[t("span",{class:"jlbinding"},"NeuralClosure.cnn")],-1)),s[129]||(s[129]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[130]||(s[130]=e(`
julia
cnn(
+)
Update parameters θ to minimize loss(dataloader(), θ) using the optimiser opt for niter iterations.
Return the a new named tuple (; opt, θ, callbackstate) with updated state and parameters.
source
`,4))]),s[176]||(s[176]=t("h2",{id:"Neural-architectures",tabindex:"-1"},[a("Neural architectures "),t("a",{class:"header-anchor",href:"#Neural-architectures","aria-label":'Permalink to "Neural architectures {#Neural-architectures}"'},"​")],-1)),s[177]||(s[177]=t("p",null,"We provide neural architectures: A convolutional neural network (CNN), group convolutional neural networks (G-CNN) and a Fourier neural operator (FNO).",-1)),t("details",x1,[t("summary",null,[s[128]||(s[128]=t("a",{id:"NeuralClosure.cnn-Tuple{}",href:"#NeuralClosure.cnn-Tuple{}"},[t("span",{class:"jlbinding"},"NeuralClosure.cnn")],-1)),s[129]||(s[129]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[130]||(s[130]=e(`
julia
cnn(
 ;
     setup,
     radii,
@@ -78,7 +78,7 @@ import{_ as r,c as n,a5 as e,j as t,a,G as i,B as d,o}from"./chunks/framework.Dr
     use_bias,
     channel_augmenter,
     rng
-)
Create CNN closure model. Return a tuple (closure, θ) where θ are the initial parameters and closure(u, θ) predicts the commutator error.
source
`,3))]),t("details",f1,[t("summary",null,[s[131]||(s[131]=t("a",{id:"NeuralClosure.GroupConv2D",href:"#NeuralClosure.GroupConv2D"},[t("span",{class:"jlbinding"},"NeuralClosure.GroupConv2D")],-1)),s[132]||(s[132]=a()),i(l,{type:"info",class:"jlObjectType jlType",text:"Type"})]),s[133]||(s[133]=e('
julia
struct GroupConv2D{C} <: LuxCore.AbstractLuxLayer
Group-equivariant convolutional layer – with respect to the p4 group. The layer is equivariant to rotations and translations of the input vector field.
The kwargs are passed to the Conv layer.
The layer has three variants:
  • If islifting then it lifts a vector input (u1, u2) into a rotation-state vector (v1, v2, v3, v4).
  • If isprojecting, it projects a rotation-state vector (u1, u2, u3, v4) into a vector (v1, v2).
  • Otherwise, it cyclically transforms the rotation-state vector (u1, u2, u3, u4) into a new rotation-state vector (v1, v2, v3, v4).
Fields
  • islifting
  • isprojecting
  • cin
  • cout
  • conv
source
',8))]),t("details",E1,[t("summary",null,[s[134]||(s[134]=t("a",{id:"NeuralClosure.gcnn-Tuple{}",href:"#NeuralClosure.gcnn-Tuple{}"},[t("span",{class:"jlbinding"},"NeuralClosure.gcnn")],-1)),s[135]||(s[135]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[136]||(s[136]=e('
julia
gcnn(; setup, radii, channels, activations, use_bias, rng)
Create CNN closure model. Return a tuple (closure, θ) where θ are the initial parameters and closure(u, θ) predicts the commutator error.
source
',3))]),t("details",w1,[t("summary",null,[s[137]||(s[137]=t("a",{id:"NeuralClosure.rot2",href:"#NeuralClosure.rot2"},[t("span",{class:"jlbinding"},"NeuralClosure.rot2")],-1)),s[138]||(s[138]=a()),i(l,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[139]||(s[139]=e('
julia
rot2(u, r)
Rotate the field u by 90 degrees counter-clockwise r - 1 times.
source
',3))]),t("details",v1,[t("summary",null,[s[140]||(s[140]=t("a",{id:"NeuralClosure.rot2-Union{Tuple{T}, Tuple{Tuple{T, T}, Any}} where T",href:"#NeuralClosure.rot2-Union{Tuple{T}, Tuple{Tuple{T, T}, Any}} where T"},[t("span",{class:"jlbinding"},"NeuralClosure.rot2")],-1)),s[141]||(s[141]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[142]||(s[142]=e('
julia
rot2(u::Tuple{T, T}, r) -> Union{Nothing, Tuple{Any, Any}}
Rotate vector fields [ux;;; uy]
source
',3))]),t("details",H1,[t("summary",null,[s[143]||(s[143]=t("a",{id:"NeuralClosure.rot2stag-Tuple{Any, Any}",href:"#NeuralClosure.rot2stag-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"NeuralClosure.rot2stag")],-1)),s[144]||(s[144]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[145]||(s[145]=e('
julia
rot2stag(u, g) -> Any
Rotate staggered grid velocity field. See also rot2.
source
',3))]),t("details",j1,[t("summary",null,[s[146]||(s[146]=t("a",{id:"NeuralClosure.vecrot2-Tuple{Any, Any}",href:"#NeuralClosure.vecrot2-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"NeuralClosure.vecrot2")],-1)),s[147]||(s[147]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[148]||(s[148]=e('
julia
vecrot2(u, r) -> Any
Rotate vector fields [ux;;; uy]
source
',3))]),t("details",C1,[t("summary",null,[s[149]||(s[149]=t("a",{id:"NeuralClosure.FourierLayer",href:"#NeuralClosure.FourierLayer"},[t("span",{class:"jlbinding"},"NeuralClosure.FourierLayer")],-1)),s[150]||(s[150]=a()),i(l,{type:"info",class:"jlObjectType jlType",text:"Type"})]),s[151]||(s[151]=e('
julia
struct FourierLayer{D, A, F} <: LuxCore.AbstractLuxLayer
Fourier layer operating on uniformly discretized functions.
Some important sizes:
  • dimension: Spatial dimension, e.g. Dimension(2) or Dimension(3).
  • (nx..., cin, nsample): Input size
  • (nx..., cout, nsample): Output size
  • nx = fill(n, dimension()): Number of points in each spatial dimension
  • n ≥ kmax: Same number of points in each spatial dimension, must be larger than cut-off wavenumber
  • kmax: Cut-off wavenumber
  • nsample: Number of input samples (treated independently)
Fields
  • dimension
  • kmax
  • cin
  • cout
  • σ
  • init_weight
source
',7))]),t("details",M1,[t("summary",null,[s[152]||(s[152]=t("a",{id:"NeuralClosure.fno-Tuple{}",href:"#NeuralClosure.fno-Tuple{}"},[t("span",{class:"jlbinding"},"NeuralClosure.fno")],-1)),s[153]||(s[153]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[154]||(s[154]=e('
julia
fno(; setup, kmax, c, σ, ψ, rng, kwargs...)
Create FNO closure model. Return a tuple (closure, θ) where θ are the initial parameters and closure(V, θ) predicts the commutator error.
source
',3))]),s[178]||(s[178]=t("h2",{id:"Data-generation",tabindex:"-1"},[a("Data generation "),t("a",{class:"header-anchor",href:"#Data-generation","aria-label":'Permalink to "Data generation {#Data-generation}"'},"​")],-1)),t("details",L1,[t("summary",null,[s[155]||(s[155]=t("a",{id:"NeuralClosure.create_io_arrays-Tuple{Any, Any}",href:"#NeuralClosure.create_io_arrays-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"NeuralClosure.create_io_arrays")],-1)),s[156]||(s[156]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[161]||(s[161]=e('
julia
create_io_arrays(data, setup) -> NamedTuple
',1)),t("p",null,[s[159]||(s[159]=a("Create ")),t("mjx-container",V1,[(o(),n("svg",N1,s[157]||(s[157]=[e('',1)]))),s[158]||(s[158]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mo",{stretchy:"false"},"("),t("mrow",{"data-mjx-texclass":"ORD"},[t("mover",null,[t("mi",null,"u"),t("mo",{stretchy:"false"},"¯")])]),t("mo",null,","),t("mi",null,"c"),t("mo",{stretchy:"false"},")")])],-1))]),s[160]||(s[160]=a(" pairs for a-priori training."))]),s[162]||(s[162]=t("p",null,[t("a",{href:"https://github.com/agdestein/IncompressibleNavierStokes.jl/blob/03acafa3566742b1f37d722d2207f5dc21d031c0/lib/NeuralClosure/src/data_generation.jl#L228",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",Z1,[t("summary",null,[s[163]||(s[163]=t("a",{id:"NeuralClosure.create_les_data-Tuple{}",href:"#NeuralClosure.create_les_data-Tuple{}"},[t("span",{class:"jlbinding"},"NeuralClosure.create_les_data")],-1)),s[164]||(s[164]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[165]||(s[165]=e(`
julia
create_les_data(
+)
Create CNN closure model. Return a tuple (closure, θ) where θ are the initial parameters and closure(u, θ) predicts the commutator error.
source
`,3))]),t("details",f1,[t("summary",null,[s[131]||(s[131]=t("a",{id:"NeuralClosure.GroupConv2D",href:"#NeuralClosure.GroupConv2D"},[t("span",{class:"jlbinding"},"NeuralClosure.GroupConv2D")],-1)),s[132]||(s[132]=a()),i(l,{type:"info",class:"jlObjectType jlType",text:"Type"})]),s[133]||(s[133]=e('
julia
struct GroupConv2D{C} <: LuxCore.AbstractLuxLayer
Group-equivariant convolutional layer – with respect to the p4 group. The layer is equivariant to rotations and translations of the input vector field.
The kwargs are passed to the Conv layer.
The layer has three variants:
  • If islifting then it lifts a vector input (u1, u2) into a rotation-state vector (v1, v2, v3, v4).
  • If isprojecting, it projects a rotation-state vector (u1, u2, u3, v4) into a vector (v1, v2).
  • Otherwise, it cyclically transforms the rotation-state vector (u1, u2, u3, u4) into a new rotation-state vector (v1, v2, v3, v4).
Fields
  • islifting
  • isprojecting
  • cin
  • cout
  • conv
source
',8))]),t("details",E1,[t("summary",null,[s[134]||(s[134]=t("a",{id:"NeuralClosure.gcnn-Tuple{}",href:"#NeuralClosure.gcnn-Tuple{}"},[t("span",{class:"jlbinding"},"NeuralClosure.gcnn")],-1)),s[135]||(s[135]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[136]||(s[136]=e('
julia
gcnn(; setup, radii, channels, activations, use_bias, rng)
Create CNN closure model. Return a tuple (closure, θ) where θ are the initial parameters and closure(u, θ) predicts the commutator error.
source
',3))]),t("details",w1,[t("summary",null,[s[137]||(s[137]=t("a",{id:"NeuralClosure.rot2",href:"#NeuralClosure.rot2"},[t("span",{class:"jlbinding"},"NeuralClosure.rot2")],-1)),s[138]||(s[138]=a()),i(l,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[139]||(s[139]=e('
julia
rot2(u, r)
Rotate the field u by 90 degrees counter-clockwise r - 1 times.
source
',3))]),t("details",v1,[t("summary",null,[s[140]||(s[140]=t("a",{id:"NeuralClosure.rot2-Union{Tuple{T}, Tuple{Tuple{T, T}, Any}} where T",href:"#NeuralClosure.rot2-Union{Tuple{T}, Tuple{Tuple{T, T}, Any}} where T"},[t("span",{class:"jlbinding"},"NeuralClosure.rot2")],-1)),s[141]||(s[141]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[142]||(s[142]=e('
julia
rot2(u::Tuple{T, T}, r) -> Union{Nothing, Tuple{Any, Any}}
Rotate vector fields [ux;;; uy]
source
',3))]),t("details",H1,[t("summary",null,[s[143]||(s[143]=t("a",{id:"NeuralClosure.rot2stag-Tuple{Any, Any}",href:"#NeuralClosure.rot2stag-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"NeuralClosure.rot2stag")],-1)),s[144]||(s[144]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[145]||(s[145]=e('
julia
rot2stag(u, g) -> Any
Rotate staggered grid velocity field. See also rot2.
source
',3))]),t("details",j1,[t("summary",null,[s[146]||(s[146]=t("a",{id:"NeuralClosure.vecrot2-Tuple{Any, Any}",href:"#NeuralClosure.vecrot2-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"NeuralClosure.vecrot2")],-1)),s[147]||(s[147]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[148]||(s[148]=e('
julia
vecrot2(u, r) -> Any
Rotate vector fields [ux;;; uy]
source
',3))]),t("details",C1,[t("summary",null,[s[149]||(s[149]=t("a",{id:"NeuralClosure.FourierLayer",href:"#NeuralClosure.FourierLayer"},[t("span",{class:"jlbinding"},"NeuralClosure.FourierLayer")],-1)),s[150]||(s[150]=a()),i(l,{type:"info",class:"jlObjectType jlType",text:"Type"})]),s[151]||(s[151]=e('
julia
struct FourierLayer{D, A, F} <: LuxCore.AbstractLuxLayer
Fourier layer operating on uniformly discretized functions.
Some important sizes:
  • dimension: Spatial dimension, e.g. Dimension(2) or Dimension(3).
  • (nx..., cin, nsample): Input size
  • (nx..., cout, nsample): Output size
  • nx = fill(n, dimension()): Number of points in each spatial dimension
  • n ≥ kmax: Same number of points in each spatial dimension, must be larger than cut-off wavenumber
  • kmax: Cut-off wavenumber
  • nsample: Number of input samples (treated independently)
Fields
  • dimension
  • kmax
  • cin
  • cout
  • σ
  • init_weight
source
',7))]),t("details",M1,[t("summary",null,[s[152]||(s[152]=t("a",{id:"NeuralClosure.fno-Tuple{}",href:"#NeuralClosure.fno-Tuple{}"},[t("span",{class:"jlbinding"},"NeuralClosure.fno")],-1)),s[153]||(s[153]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[154]||(s[154]=e('
julia
fno(; setup, kmax, c, σ, ψ, rng, kwargs...)
Create FNO closure model. Return a tuple (closure, θ) where θ are the initial parameters and closure(V, θ) predicts the commutator error.
source
',3))]),s[178]||(s[178]=t("h2",{id:"Data-generation",tabindex:"-1"},[a("Data generation "),t("a",{class:"header-anchor",href:"#Data-generation","aria-label":'Permalink to "Data generation {#Data-generation}"'},"​")],-1)),t("details",L1,[t("summary",null,[s[155]||(s[155]=t("a",{id:"NeuralClosure.create_io_arrays-Tuple{Any, Any}",href:"#NeuralClosure.create_io_arrays-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"NeuralClosure.create_io_arrays")],-1)),s[156]||(s[156]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[161]||(s[161]=e('
julia
create_io_arrays(data, setup) -> NamedTuple
',1)),t("p",null,[s[159]||(s[159]=a("Create ")),t("mjx-container",V1,[(o(),n("svg",N1,s[157]||(s[157]=[e('',1)]))),s[158]||(s[158]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mo",{stretchy:"false"},"("),t("mrow",{"data-mjx-texclass":"ORD"},[t("mover",null,[t("mi",null,"u"),t("mo",{stretchy:"false"},"¯")])]),t("mo",null,","),t("mi",null,"c"),t("mo",{stretchy:"false"},")")])],-1))]),s[160]||(s[160]=a(" pairs for a-priori training."))]),s[162]||(s[162]=t("p",null,[t("a",{href:"https://github.com/agdestein/IncompressibleNavierStokes.jl/blob/a63d20e3a410f14ba6cc612a27e89f8974019d9f/lib/NeuralClosure/src/data_generation.jl#L228",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",Z1,[t("summary",null,[s[163]||(s[163]=t("a",{id:"NeuralClosure.create_les_data-Tuple{}",href:"#NeuralClosure.create_les_data-Tuple{}"},[t("span",{class:"jlbinding"},"NeuralClosure.create_les_data")],-1)),s[164]||(s[164]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[165]||(s[165]=e(`
julia
create_les_data(
 ;
     D,
     Re,
@@ -98,7 +98,7 @@ import{_ as r,c as n,a5 as e,j as t,a,G as i,B as d,o}from"./chunks/framework.Dr
     rng,
     filenames,
     kwargs...
-)
Create filtered DNS data.
source
`,3))]),t("details",D1,[t("summary",null,[s[166]||(s[166]=t("a",{id:"NeuralClosure.filtersaver-NTuple{6, Any}",href:"#NeuralClosure.filtersaver-NTuple{6, Any}"},[t("span",{class:"jlbinding"},"NeuralClosure.filtersaver")],-1)),s[167]||(s[167]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[168]||(s[168]=e(`
julia
filtersaver(
+)
Create filtered DNS data.
source
`,3))]),t("details",D1,[t("summary",null,[s[166]||(s[166]=t("a",{id:"NeuralClosure.filtersaver-NTuple{6, Any}",href:"#NeuralClosure.filtersaver-NTuple{6, Any}"},[t("span",{class:"jlbinding"},"NeuralClosure.filtersaver")],-1)),s[167]||(s[167]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[168]||(s[168]=e(`
julia
filtersaver(
     dns,
     les,
     filters,
@@ -109,4 +109,4 @@ import{_ as r,c as n,a5 as e,j as t,a,G as i,B as d,o}from"./chunks/framework.Dr
     filenames,
     F,
     p
-)
Save filtered DNS data.
source
`,3))])])}const G1=r(p,[["render",F1]]);export{z1 as __pageData,G1 as default};
+)
Save filtered DNS data.
source
`,3))])])}const G1=r(p,[["render",F1]]);export{z1 as __pageData,G1 as default};
diff --git a/dev/assets/manual_closure.md.Bl_bxqy7.js b/dev/assets/manual_closure.md.Bp4YxggZ.lean.js
similarity index 97%
rename from dev/assets/manual_closure.md.Bl_bxqy7.js
rename to dev/assets/manual_closure.md.Bp4YxggZ.lean.js
index 4aeb55510..5a50c5546 100644
--- a/dev/assets/manual_closure.md.Bl_bxqy7.js
+++ b/dev/assets/manual_closure.md.Bp4YxggZ.lean.js
@@ -1,8 +1,8 @@
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Neural closure models ​
NeuralClosure
These features are experimental, and require cloning IncompressibleNavierStokes from GitHub:
sh
git clone https://github.com/agdestein/IncompressibleNavierStokes.jl
-cd IncompressibleNavierStokes/lib/NeuralClosure
Large eddy simulation, a closure model is required. With IncompressibleNavierStokes, a neural closure model can be trained on filtered DNS data. The discrete DNS equations are given by
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julia
collocate(u) -> Any
Interpolate velocity components to volume centers.
source
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julia
create_closure(layers...; rng)
Create neural closure model from layers.
source
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julia
create_tensorclosure(layers...; setup, rng)
Create tensor basis closure.
source
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julia
decollocate(u) -> Any
Interpolate closure force from volume centers to volume faces.
source
',3))]),t("details",q,[t("summary",null,[s[56]||(s[56]=t("a",{id:"NeuralClosure.wrappedclosure-Tuple{Any, Any}",href:"#NeuralClosure.wrappedclosure-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"NeuralClosure.wrappedclosure")],-1)),s[57]||(s[57]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[58]||(s[58]=e(`
julia
wrappedclosure(
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Neural closure models ​
NeuralClosure
These features are experimental, and require cloning IncompressibleNavierStokes from GitHub:
sh
git clone https://github.com/agdestein/IncompressibleNavierStokes.jl
+cd IncompressibleNavierStokes/lib/NeuralClosure
Large eddy simulation, a closure model is required. With IncompressibleNavierStokes, a neural closure model can be trained on filtered DNS data. The discrete DNS equations are given by
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We here assumed that ")),t("mjx-container",j,[(o(),n("svg",C,s[19]||(s[19]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D440",d:"M289 629Q289 635 232 637Q208 637 201 638T194 648Q194 649 196 659Q197 662 198 666T199 671T201 676T203 679T207 681T212 683T220 683T232 684Q238 684 262 684T307 683Q386 683 398 683T414 678Q415 674 451 396L487 117L510 154Q534 190 574 254T662 394Q837 673 839 675Q840 676 842 678T846 681L852 683H948Q965 683 988 683T1017 684Q1051 684 1051 673Q1051 668 1048 656T1045 643Q1041 637 1008 637Q968 636 957 634T939 623Q936 618 867 340T797 59Q797 55 798 54T805 50T822 48T855 46H886Q892 37 892 35Q892 19 885 5Q880 0 869 0Q864 0 828 1T736 2Q675 2 644 2T609 1Q592 1 592 11Q592 13 594 25Q598 41 602 43T625 46Q652 46 685 49Q699 52 704 61Q706 65 742 207T813 490T848 631L654 322Q458 10 453 5Q451 4 449 3Q444 0 433 0Q418 0 415 7Q413 11 374 317L335 624L267 354Q200 88 200 79Q206 46 272 46H282Q288 41 289 37T286 19Q282 3 278 1Q274 0 267 0Q265 0 255 0T221 1T157 2Q127 2 95 1T58 0Q43 0 39 2T35 11Q35 13 38 25T43 40Q45 46 65 46Q135 46 154 86Q158 92 223 354T289 629Z",style:{"stroke-width":"3"}})])])],-1)]))),s[20]||(s[20]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"M")])],-1))]),s[34]||(s[34]=a(" and ")),t("mjx-container",M,[(o(),n("svg",L,s[21]||(s[21]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"3A6",d:"M312 622Q310 623 307 625T303 629T297 631T286 634T270 635T246 636T211 637H184V683H196Q220 680 361 680T526 683H538V637H511Q468 637 447 635T422 631T411 622V533L425 531Q525 519 595 466T665 342Q665 301 642 267T583 209T506 172T425 152L411 150V61Q417 55 421 53T447 48T511 46H538V0H526Q502 3 361 3T196 0H184V46H211Q231 46 245 46T270 47T286 48T297 51T303 54T307 57T312 61V150H310Q309 151 289 153T232 166T160 195Q149 201 136 210T103 238T69 284T56 342Q56 414 128 467T294 530Q309 532 310 533H312V622ZM170 342Q170 207 307 188H312V495H309Q301 495 282 491T231 469T186 423Q170 389 170 342ZM415 188Q487 199 519 236T551 342Q551 384 539 414T507 459T470 481T434 491T415 495H410V188H415Z",style:{"stroke-width":"3"}})])])],-1)]))),s[22]||(s[22]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",{mathvariant:"normal"},"Φ")])],-1))]),s[35]||(s[35]=a(" commute, which is the case for face averaging filters. Replacing ")),t("mjx-container",V,[(o(),n("svg",N,s[23]||(s[23]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D450",d:"M34 159Q34 268 120 355T306 442Q362 442 394 418T427 355Q427 326 408 306T360 285Q341 285 330 295T319 325T330 359T352 380T366 386H367Q367 388 361 392T340 400T306 404Q276 404 249 390Q228 381 206 359Q162 315 142 235T121 119Q121 73 147 50Q169 26 205 26H209Q321 26 394 111Q403 121 406 121Q410 121 419 112T429 98T420 83T391 55T346 25T282 0T202 -11Q127 -11 81 37T34 159Z",style:{"stroke-width":"3"}})])])],-1)]))),s[24]||(s[24]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"c")])],-1))]),s[36]||(s[36]=a(" with a parameterized closure model ")),t("mjx-container",Z,[(o(),n("svg",D,s[25]||(s[25]=[e('',1)]))),s[26]||(s[26]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"m"),t("mo",{stretchy:"false"},"("),t("mrow",{"data-mjx-texclass":"ORD"},[t("mover",null,[t("mi",null,"u"),t("mo",{stretchy:"false"},"¯")])]),t("mo",null,","),t("mi",null,"θ"),t("mo",{stretchy:"false"},")"),t("mo",null,"≈"),t("mi",null,"c")])],-1))]),s[37]||(s[37]=a(" gives the LES equations for the approximate large scale velocity ")),t("mjx-container",F,[(o(),n("svg",A,s[27]||(s[27]=[e('',1)]))),s[28]||(s[28]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mover",null,[t("mi",null,"v"),t("mo",{stretchy:"false"},"¯")])]),t("mo",null,"≈"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mover",null,[t("mi",null,"u"),t("mo",{stretchy:"false"},"¯")])])])],-1))])]),t("mjx-container",O,[(o(),n("svg",S,s[38]||(s[38]=[e('',1)]))),s[39]||(s[39]=t("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[t("mtable",{displaystyle:"true",columnalign:"right 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tools.",-1)),s[43]||(s[43]=t("p",null,[t("a",{href:"https://github.com/agdestein/IncompressibleNavierStokes.jl/blob/a63d20e3a410f14ba6cc612a27e89f8974019d9f/lib/NeuralClosure/src/NeuralClosure.jl#L1-L3",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",R,[t("summary",null,[s[44]||(s[44]=t("a",{id:"NeuralClosure.collocate-Tuple{Any}",href:"#NeuralClosure.collocate-Tuple{Any}"},[t("span",{class:"jlbinding"},"NeuralClosure.collocate")],-1)),s[45]||(s[45]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[46]||(s[46]=e('
julia
collocate(u) -> Any
Interpolate velocity components to volume centers.
source
',3))]),t("details",I,[t("summary",null,[s[47]||(s[47]=t("a",{id:"NeuralClosure.create_closure-Tuple",href:"#NeuralClosure.create_closure-Tuple"},[t("span",{class:"jlbinding"},"NeuralClosure.create_closure")],-1)),s[48]||(s[48]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[49]||(s[49]=e('
julia
create_closure(layers...; rng)
Create neural closure model from layers.
source
',3))]),t("details",z,[t("summary",null,[s[50]||(s[50]=t("a",{id:"NeuralClosure.create_tensorclosure-Tuple",href:"#NeuralClosure.create_tensorclosure-Tuple"},[t("span",{class:"jlbinding"},"NeuralClosure.create_tensorclosure")],-1)),s[51]||(s[51]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[52]||(s[52]=e('
julia
create_tensorclosure(layers...; setup, rng)
Create tensor basis closure.
source
',3))]),t("details",G,[t("summary",null,[s[53]||(s[53]=t("a",{id:"NeuralClosure.decollocate-Tuple{Any}",href:"#NeuralClosure.decollocate-Tuple{Any}"},[t("span",{class:"jlbinding"},"NeuralClosure.decollocate")],-1)),s[54]||(s[54]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[55]||(s[55]=e('
julia
decollocate(u) -> Any
Interpolate closure force from volume centers to volume faces.
source
',3))]),t("details",q,[t("summary",null,[s[56]||(s[56]=t("a",{id:"NeuralClosure.wrappedclosure-Tuple{Any, Any}",href:"#NeuralClosure.wrappedclosure-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"NeuralClosure.wrappedclosure")],-1)),s[57]||(s[57]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[58]||(s[58]=e(`
julia
wrappedclosure(
     m,
     setup
-) -> NeuralClosure.var"#neuralclosure#1"
Wrap closure model and parameters so that it can be used in the solver.
source
`,3))]),s[172]||(s[172]=t("h2",{id:"filters",tabindex:"-1"},[a("Filters "),t("a",{class:"header-anchor",href:"#filters","aria-label":'Permalink to "Filters"'},"​")],-1)),s[173]||(s[173]=t("p",null,"The following filters are available:",-1)),t("details",X,[t("summary",null,[s[59]||(s[59]=t("a",{id:"NeuralClosure.AbstractFilter",href:"#NeuralClosure.AbstractFilter"},[t("span",{class:"jlbinding"},"NeuralClosure.AbstractFilter")],-1)),s[60]||(s[60]=a()),i(l,{type:"info",class:"jlObjectType jlType",text:"Type"})]),s[61]||(s[61]=e('
julia
abstract type AbstractFilter
Discrete DNS filter.
Subtypes ConcreteFilter should implement the in-place method:
(::ConcreteFilter)(v, u, setup_les, compression)
which filters the DNS field u and put result in LES field v. Then the out-of place method:
(::ConcreteFilter)(u, setup_les, compression)
automatically becomes available.
Fields
source
',9))]),t("details",J,[t("summary",null,[s[62]||(s[62]=t("a",{id:"NeuralClosure.FaceAverage",href:"#NeuralClosure.FaceAverage"},[t("span",{class:"jlbinding"},"NeuralClosure.FaceAverage")],-1)),s[63]||(s[63]=a()),i(l,{type:"info",class:"jlObjectType jlType",text:"Type"})]),s[64]||(s[64]=e('
julia
struct FaceAverage <: NeuralClosure.AbstractFilter
Average fine grid velocity field over coarse volume face.
Fields
source
',4))]),t("details",P,[t("summary",null,[s[65]||(s[65]=t("a",{id:"NeuralClosure.VolumeAverage",href:"#NeuralClosure.VolumeAverage"},[t("span",{class:"jlbinding"},"NeuralClosure.VolumeAverage")],-1)),s[66]||(s[66]=a()),i(l,{type:"info",class:"jlObjectType jlType",text:"Type"})]),s[67]||(s[67]=e('
julia
struct VolumeAverage <: NeuralClosure.AbstractFilter
Average fine grid velocity field over coarse volume.
Fields
source
',4))]),t("details",U,[t("summary",null,[s[68]||(s[68]=t("a",{id:"NeuralClosure.reconstruct!-NTuple{5, Any}",href:"#NeuralClosure.reconstruct!-NTuple{5, Any}"},[t("span",{class:"jlbinding"},"NeuralClosure.reconstruct!")],-1)),s[69]||(s[69]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[70]||(s[70]=e('
julia
reconstruct!(u, v, setup_dns, setup_les, comp) -> Any
Reconstruct DNS velocity u from LES velocity v.
source
',3))]),t("details",W,[t("summary",null,[s[71]||(s[71]=t("a",{id:"NeuralClosure.reconstruct-NTuple{4, Any}",href:"#NeuralClosure.reconstruct-NTuple{4, Any}"},[t("span",{class:"jlbinding"},"NeuralClosure.reconstruct")],-1)),s[72]||(s[72]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[73]||(s[73]=e('
julia
reconstruct(v, setup_dns, setup_les, comp) -> Any
Reconstruct DNS velocity field. See also reconstruct!.
source
',3))]),s[174]||(s[174]=t("h2",{id:"training",tabindex:"-1"},[a("Training "),t("a",{class:"header-anchor",href:"#training","aria-label":'Permalink to "Training"'},"​")],-1)),t("p",null,[s[78]||(s[78]=a("To improve the model parameters, we exploit exact filtered DNS data ")),t("mjx-container",$,[(o(),n("svg",K,s[74]||(s[74]=[e('',1)]))),s[75]||(s[75]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mover",null,[t("mi",null,"u"),t("mo",{stretchy:"false"},"¯")])])])],-1))]),s[79]||(s[79]=a(" and exact commutator errors ")),t("mjx-container",Y,[(o(),n("svg",_,s[76]||(s[76]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D450",d:"M34 159Q34 268 120 355T306 442Q362 442 394 418T427 355Q427 326 408 306T360 285Q341 285 330 295T319 325T330 359T352 380T366 386H367Q367 388 361 392T340 400T306 404Q276 404 249 390Q228 381 206 359Q162 315 142 235T121 119Q121 73 147 50Q169 26 205 26H209Q321 26 394 111Q403 121 406 121Q410 121 419 112T429 98T420 83T391 55T346 25T282 0T202 -11Q127 -11 81 37T34 159Z",style:{"stroke-width":"3"}})])])],-1)]))),s[77]||(s[77]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"c")])],-1))]),s[80]||(s[80]=a(" obtained through DNS. The model is trained by minimizing the a priori loss function"))]),t("mjx-container",t1,[(o(),n("svg",s1,s[81]||(s[81]=[e('',1)]))),s[82]||(s[82]=t("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[t("msup",null,[t("mi",null,"L"),t("mtext",null,"prior")]),t("mo",{stretchy:"false"},"("),t("mi",null,"θ"),t("mo",{stretchy:"false"},")"),t("mo",null,"="),t("mo",{"data-mjx-texclass":"ORD"},"∥"),t("mi",null,"m"),t("mo",{stretchy:"false"},"("),t("mrow",{"data-mjx-texclass":"ORD"},[t("mover",null,[t("mi",null,"u"),t("mo",{stretchy:"false"},"¯")])]),t("mo",null,","),t("mi",null,"θ"),t("mo",{stretchy:"false"},")"),t("mo",null,"−"),t("mi",null,"c"),t("msup",null,[t("mo",{"data-mjx-texclass":"ORD"},"∥"),t("mn",null,"2")]),t("mo",null,",")])],-1))]),s[175]||(s[175]=t("p",null,"or the a posteriori loss function",-1)),t("mjx-container",a1,[(o(),n("svg",e1,s[83]||(s[83]=[e('',1)]))),s[84]||(s[84]=t("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[t("msup",null,[t("mi",null,"L"),t("mtext",null,"post")]),t("mo",{stretchy:"false"},"("),t("mi",null,"θ"),t("mo",{stretchy:"false"},")"),t("mo",null,"="),t("mo",{"data-mjx-texclass":"ORD"},"∥"),t("msub",null,[t("mrow",{"data-mjx-texclass":"ORD"},[t("mover",null,[t("mi",null,"v"),t("mo",{stretchy:"false"},"¯")])]),t("mi",null,"θ")]),t("mo",null,"−"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mover",null,[t("mi",null,"u"),t("mo",{stretchy:"false"},"¯")])]),t("msup",null,[t("mo",{"data-mjx-texclass":"ORD"},"∥"),t("mn",null,"2")]),t("mo",null,",")])],-1))]),t("p",null,[s[91]||(s[91]=a("where ")),t("mjx-container",l1,[(o(),n("svg",i1,s[85]||(s[85]=[e('',1)]))),s[86]||(s[86]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mrow",{"data-mjx-texclass":"ORD"},[t("mover",null,[t("mi",null,"v"),t("mo",{stretchy:"false"},"¯")])]),t("mi",null,"θ")])])],-1))]),s[92]||(s[92]=a(" is the solution to the LES equation for the given parameters ")),t("mjx-container",n1,[(o(),n("svg",o1,s[87]||(s[87]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D703",d:"M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z",style:{"stroke-width":"3"}})])])],-1)]))),s[88]||(s[88]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"θ")])],-1))]),s[93]||(s[93]=a(". The prior loss is easy to evaluate and easy to differentiate, as it does not involve solving the ODE. However, minimizing ")),t("mjx-container",r1,[(o(),n("svg",d1,s[89]||(s[89]=[e('',1)]))),s[90]||(s[90]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msup",null,[t("mi",null,"L"),t("mtext",null,"prior")])])],-1))]),s[94]||(s[94]=a(" does not take into account the effect of the prediction error on the LES solution error. The posterior loss does, but has a longer computational chain involving solving the LES ODE."))]),t("details",p1,[t("summary",null,[s[95]||(s[95]=t("a",{id:"NeuralClosure.create_callback-Tuple{Any}",href:"#NeuralClosure.create_callback-Tuple{Any}"},[t("span",{class:"jlbinding"},"NeuralClosure.create_callback")],-1)),s[96]||(s[96]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[97]||(s[97]=e(`
julia
create_callback(
+) -> NeuralClosure.var"#neuralclosure#1"
Wrap closure model and parameters so that it can be used in the solver.
source
`,3))]),s[172]||(s[172]=t("h2",{id:"filters",tabindex:"-1"},[a("Filters "),t("a",{class:"header-anchor",href:"#filters","aria-label":'Permalink to "Filters"'},"​")],-1)),s[173]||(s[173]=t("p",null,"The following filters are available:",-1)),t("details",X,[t("summary",null,[s[59]||(s[59]=t("a",{id:"NeuralClosure.AbstractFilter",href:"#NeuralClosure.AbstractFilter"},[t("span",{class:"jlbinding"},"NeuralClosure.AbstractFilter")],-1)),s[60]||(s[60]=a()),i(l,{type:"info",class:"jlObjectType jlType",text:"Type"})]),s[61]||(s[61]=e('
julia
abstract type AbstractFilter
Discrete DNS filter.
Subtypes ConcreteFilter should implement the in-place method:
(::ConcreteFilter)(v, u, setup_les, compression)
which filters the DNS field u and put result in LES field v. Then the out-of place method:
(::ConcreteFilter)(u, setup_les, compression)
automatically becomes available.
Fields
source
',9))]),t("details",J,[t("summary",null,[s[62]||(s[62]=t("a",{id:"NeuralClosure.FaceAverage",href:"#NeuralClosure.FaceAverage"},[t("span",{class:"jlbinding"},"NeuralClosure.FaceAverage")],-1)),s[63]||(s[63]=a()),i(l,{type:"info",class:"jlObjectType jlType",text:"Type"})]),s[64]||(s[64]=e('
julia
struct FaceAverage <: NeuralClosure.AbstractFilter
Average fine grid velocity field over coarse volume face.
Fields
source
',4))]),t("details",P,[t("summary",null,[s[65]||(s[65]=t("a",{id:"NeuralClosure.VolumeAverage",href:"#NeuralClosure.VolumeAverage"},[t("span",{class:"jlbinding"},"NeuralClosure.VolumeAverage")],-1)),s[66]||(s[66]=a()),i(l,{type:"info",class:"jlObjectType jlType",text:"Type"})]),s[67]||(s[67]=e('
julia
struct VolumeAverage <: NeuralClosure.AbstractFilter
Average fine grid velocity field over coarse volume.
Fields
source
',4))]),t("details",U,[t("summary",null,[s[68]||(s[68]=t("a",{id:"NeuralClosure.reconstruct!-NTuple{5, Any}",href:"#NeuralClosure.reconstruct!-NTuple{5, Any}"},[t("span",{class:"jlbinding"},"NeuralClosure.reconstruct!")],-1)),s[69]||(s[69]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[70]||(s[70]=e('
julia
reconstruct!(u, v, setup_dns, setup_les, comp) -> Any
Reconstruct DNS velocity u from LES velocity v.
source
',3))]),t("details",W,[t("summary",null,[s[71]||(s[71]=t("a",{id:"NeuralClosure.reconstruct-NTuple{4, Any}",href:"#NeuralClosure.reconstruct-NTuple{4, Any}"},[t("span",{class:"jlbinding"},"NeuralClosure.reconstruct")],-1)),s[72]||(s[72]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[73]||(s[73]=e('
julia
reconstruct(v, setup_dns, setup_les, comp) -> Any
Reconstruct DNS velocity field. See also reconstruct!.
source
',3))]),s[174]||(s[174]=t("h2",{id:"training",tabindex:"-1"},[a("Training "),t("a",{class:"header-anchor",href:"#training","aria-label":'Permalink to "Training"'},"​")],-1)),t("p",null,[s[78]||(s[78]=a("To improve the model parameters, we exploit exact filtered DNS data ")),t("mjx-container",$,[(o(),n("svg",K,s[74]||(s[74]=[e('',1)]))),s[75]||(s[75]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mover",null,[t("mi",null,"u"),t("mo",{stretchy:"false"},"¯")])])])],-1))]),s[79]||(s[79]=a(" and exact commutator errors ")),t("mjx-container",Y,[(o(),n("svg",_,s[76]||(s[76]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D450",d:"M34 159Q34 268 120 355T306 442Q362 442 394 418T427 355Q427 326 408 306T360 285Q341 285 330 295T319 325T330 359T352 380T366 386H367Q367 388 361 392T340 400T306 404Q276 404 249 390Q228 381 206 359Q162 315 142 235T121 119Q121 73 147 50Q169 26 205 26H209Q321 26 394 111Q403 121 406 121Q410 121 419 112T429 98T420 83T391 55T346 25T282 0T202 -11Q127 -11 81 37T34 159Z",style:{"stroke-width":"3"}})])])],-1)]))),s[77]||(s[77]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"c")])],-1))]),s[80]||(s[80]=a(" obtained through DNS. The model is trained by minimizing the a priori loss function"))]),t("mjx-container",t1,[(o(),n("svg",s1,s[81]||(s[81]=[e('',1)]))),s[82]||(s[82]=t("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[t("msup",null,[t("mi",null,"L"),t("mtext",null,"prior")]),t("mo",{stretchy:"false"},"("),t("mi",null,"θ"),t("mo",{stretchy:"false"},")"),t("mo",null,"="),t("mo",{"data-mjx-texclass":"ORD"},"∥"),t("mi",null,"m"),t("mo",{stretchy:"false"},"("),t("mrow",{"data-mjx-texclass":"ORD"},[t("mover",null,[t("mi",null,"u"),t("mo",{stretchy:"false"},"¯")])]),t("mo",null,","),t("mi",null,"θ"),t("mo",{stretchy:"false"},")"),t("mo",null,"−"),t("mi",null,"c"),t("msup",null,[t("mo",{"data-mjx-texclass":"ORD"},"∥"),t("mn",null,"2")]),t("mo",null,",")])],-1))]),s[175]||(s[175]=t("p",null,"or the a posteriori loss function",-1)),t("mjx-container",a1,[(o(),n("svg",e1,s[83]||(s[83]=[e('',1)]))),s[84]||(s[84]=t("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[t("msup",null,[t("mi",null,"L"),t("mtext",null,"post")]),t("mo",{stretchy:"false"},"("),t("mi",null,"θ"),t("mo",{stretchy:"false"},")"),t("mo",null,"="),t("mo",{"data-mjx-texclass":"ORD"},"∥"),t("msub",null,[t("mrow",{"data-mjx-texclass":"ORD"},[t("mover",null,[t("mi",null,"v"),t("mo",{stretchy:"false"},"¯")])]),t("mi",null,"θ")]),t("mo",null,"−"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mover",null,[t("mi",null,"u"),t("mo",{stretchy:"false"},"¯")])]),t("msup",null,[t("mo",{"data-mjx-texclass":"ORD"},"∥"),t("mn",null,"2")]),t("mo",null,",")])],-1))]),t("p",null,[s[91]||(s[91]=a("where ")),t("mjx-container",l1,[(o(),n("svg",i1,s[85]||(s[85]=[e('',1)]))),s[86]||(s[86]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mrow",{"data-mjx-texclass":"ORD"},[t("mover",null,[t("mi",null,"v"),t("mo",{stretchy:"false"},"¯")])]),t("mi",null,"θ")])])],-1))]),s[92]||(s[92]=a(" is the solution to the LES equation for the given parameters ")),t("mjx-container",n1,[(o(),n("svg",o1,s[87]||(s[87]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D703",d:"M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z",style:{"stroke-width":"3"}})])])],-1)]))),s[88]||(s[88]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"θ")])],-1))]),s[93]||(s[93]=a(". The prior loss is easy to evaluate and easy to differentiate, as it does not involve solving the ODE. However, minimizing ")),t("mjx-container",r1,[(o(),n("svg",d1,s[89]||(s[89]=[e('',1)]))),s[90]||(s[90]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msup",null,[t("mi",null,"L"),t("mtext",null,"prior")])])],-1))]),s[94]||(s[94]=a(" does not take into account the effect of the prediction error on the LES solution error. The posterior loss does, but has a longer computational chain involving solving the LES ODE."))]),t("details",p1,[t("summary",null,[s[95]||(s[95]=t("a",{id:"NeuralClosure.create_callback-Tuple{Any}",href:"#NeuralClosure.create_callback-Tuple{Any}"},[t("span",{class:"jlbinding"},"NeuralClosure.create_callback")],-1)),s[96]||(s[96]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[97]||(s[97]=e(`
julia
create_callback(
     err;
     θ,
     callbackstate,
@@ -11,29 +11,29 @@ import{_ as r,c as n,a5 as e,j as t,a,G as i,B as d,o}from"./chunks/framework.Dr
     displayupdates,
     figfile,
     nupdate
-)
Create convergence plot for relative error between f(x, θ) and y. At each callback, plot is updated and current error is printed.
If state is nonempty, it also plots previous convergence.
If not using interactive GLMakie window, set displayupdates to true.
source
`,5))]),t("details",T1,[t("summary",null,[s[98]||(s[98]=t("a",{id:"NeuralClosure.create_dataloader_post-Tuple{Any}",href:"#NeuralClosure.create_dataloader_post-Tuple{Any}"},[t("span",{class:"jlbinding"},"NeuralClosure.create_dataloader_post")],-1)),s[99]||(s[99]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[100]||(s[100]=e(`
julia
create_dataloader_post(
+)
Create convergence plot for relative error between f(x, θ) and y. At each callback, plot is updated and current error is printed.
If state is nonempty, it also plots previous convergence.
If not using interactive GLMakie window, set displayupdates to true.
source
`,5))]),t("details",T1,[t("summary",null,[s[98]||(s[98]=t("a",{id:"NeuralClosure.create_dataloader_post-Tuple{Any}",href:"#NeuralClosure.create_dataloader_post-Tuple{Any}"},[t("span",{class:"jlbinding"},"NeuralClosure.create_dataloader_post")],-1)),s[99]||(s[99]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[100]||(s[100]=e(`
julia
create_dataloader_post(
     trajectories;
     ntrajectory,
     nunroll,
     device
-)
Create trajectory dataloader.
source
`,3))]),t("details",Q1,[t("summary",null,[s[101]||(s[101]=t("a",{id:"NeuralClosure.create_dataloader_prior-Tuple{Any}",href:"#NeuralClosure.create_dataloader_prior-Tuple{Any}"},[t("span",{class:"jlbinding"},"NeuralClosure.create_dataloader_prior")],-1)),s[102]||(s[102]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[103]||(s[103]=e(`
julia
create_dataloader_prior(
+)
Create trajectory dataloader.
source
`,3))]),t("details",Q1,[t("summary",null,[s[101]||(s[101]=t("a",{id:"NeuralClosure.create_dataloader_prior-Tuple{Any}",href:"#NeuralClosure.create_dataloader_prior-Tuple{Any}"},[t("span",{class:"jlbinding"},"NeuralClosure.create_dataloader_prior")],-1)),s[102]||(s[102]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[103]||(s[103]=e(`
julia
create_dataloader_prior(
     data;
     batchsize,
     device
-) -> NeuralClosure.var"#dataloader#44"{Int64}
Create dataloader that uses a batch of batchsize random samples from data at each evaluation. The batch is moved to device.
source
`,3))]),t("details",m1,[t("summary",null,[s[104]||(s[104]=t("a",{id:"NeuralClosure.create_loss_post-Tuple{}",href:"#NeuralClosure.create_loss_post-Tuple{}"},[t("span",{class:"jlbinding"},"NeuralClosure.create_loss_post")],-1)),s[105]||(s[105]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[106]||(s[106]=e(`
julia
create_loss_post(
+) -> NeuralClosure.var"#dataloader#44"{Int64}
Create dataloader that uses a batch of batchsize random samples from data at each evaluation. The batch is moved to device.
source
`,3))]),t("details",m1,[t("summary",null,[s[104]||(s[104]=t("a",{id:"NeuralClosure.create_loss_post-Tuple{}",href:"#NeuralClosure.create_loss_post-Tuple{}"},[t("span",{class:"jlbinding"},"NeuralClosure.create_loss_post")],-1)),s[105]||(s[105]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[106]||(s[106]=e(`
julia
create_loss_post(
 ;
     setup,
     method,
     psolver,
     closure,
     nsubstep
-)
Create a-posteriori loss function.
source
`,3))]),t("details",h1,[t("summary",null,[s[107]||(s[107]=t("a",{id:"NeuralClosure.create_loss_prior",href:"#NeuralClosure.create_loss_prior"},[t("span",{class:"jlbinding"},"NeuralClosure.create_loss_prior")],-1)),s[108]||(s[108]=a()),i(l,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[109]||(s[109]=e(`
julia
create_loss_prior(
+)
Create a-posteriori loss function.
source
`,3))]),t("details",h1,[t("summary",null,[s[107]||(s[107]=t("a",{id:"NeuralClosure.create_loss_prior",href:"#NeuralClosure.create_loss_prior"},[t("span",{class:"jlbinding"},"NeuralClosure.create_loss_prior")],-1)),s[108]||(s[108]=a()),i(l,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[109]||(s[109]=e(`
julia
create_loss_prior(
     f
 ) -> NeuralClosure.var"#loss_prior#57"{_A, NeuralClosure.var"#55#56"} where _A
 create_loss_prior(
     f,
     normalize
-) -> NeuralClosure.var"#loss_prior#57"
Return mean squared error loss for the predictor f.
source
`,3))]),t("details",u1,[t("summary",null,[s[110]||(s[110]=t("a",{id:"NeuralClosure.create_relerr_post-Tuple{}",href:"#NeuralClosure.create_relerr_post-Tuple{}"},[t("span",{class:"jlbinding"},"NeuralClosure.create_relerr_post")],-1)),s[111]||(s[111]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[112]||(s[112]=e(`
julia
create_relerr_post(
+) -> NeuralClosure.var"#loss_prior#57"
Return mean squared error loss for the predictor f.
source
`,3))]),t("details",u1,[t("summary",null,[s[110]||(s[110]=t("a",{id:"NeuralClosure.create_relerr_post-Tuple{}",href:"#NeuralClosure.create_relerr_post-Tuple{}"},[t("span",{class:"jlbinding"},"NeuralClosure.create_relerr_post")],-1)),s[111]||(s[111]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[112]||(s[112]=e(`
julia
create_relerr_post(
 ;
     data,
     setup,
@@ -41,7 +41,7 @@ import{_ as r,c as n,a5 as e,j as t,a,G as i,B as d,o}from"./chunks/framework.Dr
     psolver,
     closure_model,
     nsubstep
-)
Create a-posteriori relative error.
source
`,3))]),t("details",c1,[t("summary",null,[s[113]||(s[113]=t("a",{id:"NeuralClosure.create_relerr_prior-Tuple{Any, Any, Any}",href:"#NeuralClosure.create_relerr_prior-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"NeuralClosure.create_relerr_prior")],-1)),s[114]||(s[114]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[115]||(s[115]=e('
julia
create_relerr_prior(f, x, y) -> NeuralClosure.var"#58#59"
Create a-priori error.
source
',3))]),t("details",g1,[t("summary",null,[s[116]||(s[116]=t("a",{id:"NeuralClosure.create_relerr_symmetry_post-Tuple{}",href:"#NeuralClosure.create_relerr_symmetry_post-Tuple{}"},[t("span",{class:"jlbinding"},"NeuralClosure.create_relerr_symmetry_post")],-1)),s[117]||(s[117]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[118]||(s[118]=e(`
julia
create_relerr_symmetry_post(
+)
Create a-posteriori relative error.
source
`,3))]),t("details",g1,[t("summary",null,[s[113]||(s[113]=t("a",{id:"NeuralClosure.create_relerr_prior-Tuple{Any, Any, Any}",href:"#NeuralClosure.create_relerr_prior-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"NeuralClosure.create_relerr_prior")],-1)),s[114]||(s[114]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[115]||(s[115]=e('
julia
create_relerr_prior(f, x, y) -> NeuralClosure.var"#58#59"
Create a-priori error.
source
',3))]),t("details",c1,[t("summary",null,[s[116]||(s[116]=t("a",{id:"NeuralClosure.create_relerr_symmetry_post-Tuple{}",href:"#NeuralClosure.create_relerr_symmetry_post-Tuple{}"},[t("span",{class:"jlbinding"},"NeuralClosure.create_relerr_symmetry_post")],-1)),s[117]||(s[117]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[118]||(s[118]=e(`
julia
create_relerr_symmetry_post(
 ;
     u,
     setup,
@@ -50,7 +50,7 @@ import{_ as r,c as n,a5 as e,j as t,a,G as i,B as d,o}from"./chunks/framework.Dr
     Δt,
     nstep,
     g
-)
Create a-posteriori symmetry error.
source
`,3))]),t("details",k1,[t("summary",null,[s[119]||(s[119]=t("a",{id:"NeuralClosure.create_relerr_symmetry_prior-Tuple{}",href:"#NeuralClosure.create_relerr_symmetry_prior-Tuple{}"},[t("span",{class:"jlbinding"},"NeuralClosure.create_relerr_symmetry_prior")],-1)),s[120]||(s[120]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[121]||(s[121]=e('
julia
create_relerr_symmetry_prior(; u, setup, g)
Create a-priori equivariance error.
source
',3))]),t("details",y1,[t("summary",null,[s[122]||(s[122]=t("a",{id:"NeuralClosure.train-Tuple{}",href:"#NeuralClosure.train-Tuple{}"},[t("span",{class:"jlbinding"},"NeuralClosure.train")],-1)),s[123]||(s[123]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[124]||(s[124]=e(`
julia
train(
+)
Create a-posteriori symmetry error.
source
`,3))]),t("details",k1,[t("summary",null,[s[119]||(s[119]=t("a",{id:"NeuralClosure.create_relerr_symmetry_prior-Tuple{}",href:"#NeuralClosure.create_relerr_symmetry_prior-Tuple{}"},[t("span",{class:"jlbinding"},"NeuralClosure.create_relerr_symmetry_prior")],-1)),s[120]||(s[120]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[121]||(s[121]=e('
julia
create_relerr_symmetry_prior(; u, setup, g)
Create a-priori equivariance error.
source
',3))]),t("details",y1,[t("summary",null,[s[122]||(s[122]=t("a",{id:"NeuralClosure.train-Tuple{}",href:"#NeuralClosure.train-Tuple{}"},[t("span",{class:"jlbinding"},"NeuralClosure.train")],-1)),s[123]||(s[123]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[124]||(s[124]=e(`
julia
train(
 ;
     dataloader,
     loss,
@@ -59,7 +59,7 @@ import{_ as r,c as n,a5 as e,j as t,a,G as i,B as d,o}from"./chunks/framework.Dr
     callback,
     callbackstate,
     λ
-)
Update parameters θ to minimize loss(dataloader(), θ) using the optimiser opt for niter iterations.
Return the a new named tuple (; opt, θ, callbackstate) with updated state and parameters.
source
`,4))]),t("details",b1,[t("summary",null,[s[125]||(s[125]=t("a",{id:"NeuralClosure.trainepoch-Tuple{}",href:"#NeuralClosure.trainepoch-Tuple{}"},[t("span",{class:"jlbinding"},"NeuralClosure.trainepoch")],-1)),s[126]||(s[126]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[127]||(s[127]=e(`
julia
trainepoch(
+)
Update parameters θ to minimize loss(dataloader(), θ) using the optimiser opt for niter iterations.
Return the a new named tuple (; opt, θ, callbackstate) with updated state and parameters.
source
`,4))]),t("details",b1,[t("summary",null,[s[125]||(s[125]=t("a",{id:"NeuralClosure.trainepoch-Tuple{}",href:"#NeuralClosure.trainepoch-Tuple{}"},[t("span",{class:"jlbinding"},"NeuralClosure.trainepoch")],-1)),s[126]||(s[126]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[127]||(s[127]=e(`
julia
trainepoch(
 ;
     dataloader,
     loss,
@@ -69,7 +69,7 @@ import{_ as r,c as n,a5 as e,j as t,a,G as i,B as d,o}from"./chunks/framework.Dr
     device,
     noiselevel,
     λ
-)
Update parameters θ to minimize loss(dataloader(), θ) using the optimiser opt for niter iterations.
Return the a new named tuple (; opt, θ, callbackstate) with updated state and parameters.
source
`,4))]),s[176]||(s[176]=t("h2",{id:"Neural-architectures",tabindex:"-1"},[a("Neural architectures "),t("a",{class:"header-anchor",href:"#Neural-architectures","aria-label":'Permalink to "Neural architectures {#Neural-architectures}"'},"​")],-1)),s[177]||(s[177]=t("p",null,"We provide neural architectures: A convolutional neural network (CNN), group convolutional neural networks (G-CNN) and a Fourier neural operator (FNO).",-1)),t("details",x1,[t("summary",null,[s[128]||(s[128]=t("a",{id:"NeuralClosure.cnn-Tuple{}",href:"#NeuralClosure.cnn-Tuple{}"},[t("span",{class:"jlbinding"},"NeuralClosure.cnn")],-1)),s[129]||(s[129]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[130]||(s[130]=e(`
julia
cnn(
+)
Update parameters θ to minimize loss(dataloader(), θ) using the optimiser opt for niter iterations.
Return the a new named tuple (; opt, θ, callbackstate) with updated state and parameters.
source
`,4))]),s[176]||(s[176]=t("h2",{id:"Neural-architectures",tabindex:"-1"},[a("Neural architectures "),t("a",{class:"header-anchor",href:"#Neural-architectures","aria-label":'Permalink to "Neural architectures {#Neural-architectures}"'},"​")],-1)),s[177]||(s[177]=t("p",null,"We provide neural architectures: A convolutional neural network (CNN), group convolutional neural networks (G-CNN) and a Fourier neural operator (FNO).",-1)),t("details",x1,[t("summary",null,[s[128]||(s[128]=t("a",{id:"NeuralClosure.cnn-Tuple{}",href:"#NeuralClosure.cnn-Tuple{}"},[t("span",{class:"jlbinding"},"NeuralClosure.cnn")],-1)),s[129]||(s[129]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[130]||(s[130]=e(`
julia
cnn(
 ;
     setup,
     radii,
@@ -78,7 +78,7 @@ import{_ as r,c as n,a5 as e,j as t,a,G as i,B as d,o}from"./chunks/framework.Dr
     use_bias,
     channel_augmenter,
     rng
-)
Create CNN closure model. Return a tuple (closure, θ) where θ are the initial parameters and closure(u, θ) predicts the commutator error.
source
`,3))]),t("details",f1,[t("summary",null,[s[131]||(s[131]=t("a",{id:"NeuralClosure.GroupConv2D",href:"#NeuralClosure.GroupConv2D"},[t("span",{class:"jlbinding"},"NeuralClosure.GroupConv2D")],-1)),s[132]||(s[132]=a()),i(l,{type:"info",class:"jlObjectType jlType",text:"Type"})]),s[133]||(s[133]=e('
julia
struct GroupConv2D{C} <: LuxCore.AbstractLuxLayer
Group-equivariant convolutional layer – with respect to the p4 group. The layer is equivariant to rotations and translations of the input vector field.
The kwargs are passed to the Conv layer.
The layer has three variants:
  • If islifting then it lifts a vector input (u1, u2) into a rotation-state vector (v1, v2, v3, v4).
  • If isprojecting, it projects a rotation-state vector (u1, u2, u3, v4) into a vector (v1, v2).
  • Otherwise, it cyclically transforms the rotation-state vector (u1, u2, u3, u4) into a new rotation-state vector (v1, v2, v3, v4).
Fields
  • islifting
  • isprojecting
  • cin
  • cout
  • conv
source
',8))]),t("details",E1,[t("summary",null,[s[134]||(s[134]=t("a",{id:"NeuralClosure.gcnn-Tuple{}",href:"#NeuralClosure.gcnn-Tuple{}"},[t("span",{class:"jlbinding"},"NeuralClosure.gcnn")],-1)),s[135]||(s[135]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[136]||(s[136]=e('
julia
gcnn(; setup, radii, channels, activations, use_bias, rng)
Create CNN closure model. Return a tuple (closure, θ) where θ are the initial parameters and closure(u, θ) predicts the commutator error.
source
',3))]),t("details",w1,[t("summary",null,[s[137]||(s[137]=t("a",{id:"NeuralClosure.rot2",href:"#NeuralClosure.rot2"},[t("span",{class:"jlbinding"},"NeuralClosure.rot2")],-1)),s[138]||(s[138]=a()),i(l,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[139]||(s[139]=e('
julia
rot2(u, r)
Rotate the field u by 90 degrees counter-clockwise r - 1 times.
source
',3))]),t("details",v1,[t("summary",null,[s[140]||(s[140]=t("a",{id:"NeuralClosure.rot2-Union{Tuple{T}, Tuple{Tuple{T, T}, Any}} where T",href:"#NeuralClosure.rot2-Union{Tuple{T}, Tuple{Tuple{T, T}, Any}} where T"},[t("span",{class:"jlbinding"},"NeuralClosure.rot2")],-1)),s[141]||(s[141]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[142]||(s[142]=e('
julia
rot2(u::Tuple{T, T}, r) -> Union{Nothing, Tuple{Any, Any}}
Rotate vector fields [ux;;; uy]
source
',3))]),t("details",H1,[t("summary",null,[s[143]||(s[143]=t("a",{id:"NeuralClosure.rot2stag-Tuple{Any, Any}",href:"#NeuralClosure.rot2stag-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"NeuralClosure.rot2stag")],-1)),s[144]||(s[144]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[145]||(s[145]=e('
julia
rot2stag(u, g) -> Any
Rotate staggered grid velocity field. See also rot2.
source
',3))]),t("details",j1,[t("summary",null,[s[146]||(s[146]=t("a",{id:"NeuralClosure.vecrot2-Tuple{Any, Any}",href:"#NeuralClosure.vecrot2-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"NeuralClosure.vecrot2")],-1)),s[147]||(s[147]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[148]||(s[148]=e('
julia
vecrot2(u, r) -> Any
Rotate vector fields [ux;;; uy]
source
',3))]),t("details",C1,[t("summary",null,[s[149]||(s[149]=t("a",{id:"NeuralClosure.FourierLayer",href:"#NeuralClosure.FourierLayer"},[t("span",{class:"jlbinding"},"NeuralClosure.FourierLayer")],-1)),s[150]||(s[150]=a()),i(l,{type:"info",class:"jlObjectType jlType",text:"Type"})]),s[151]||(s[151]=e('
julia
struct FourierLayer{D, A, F} <: LuxCore.AbstractLuxLayer
Fourier layer operating on uniformly discretized functions.
Some important sizes:
  • dimension: Spatial dimension, e.g. Dimension(2) or Dimension(3).
  • (nx..., cin, nsample): Input size
  • (nx..., cout, nsample): Output size
  • nx = fill(n, dimension()): Number of points in each spatial dimension
  • n ≥ kmax: Same number of points in each spatial dimension, must be larger than cut-off wavenumber
  • kmax: Cut-off wavenumber
  • nsample: Number of input samples (treated independently)
Fields
  • dimension
  • kmax
  • cin
  • cout
  • σ
  • init_weight
source
',7))]),t("details",M1,[t("summary",null,[s[152]||(s[152]=t("a",{id:"NeuralClosure.fno-Tuple{}",href:"#NeuralClosure.fno-Tuple{}"},[t("span",{class:"jlbinding"},"NeuralClosure.fno")],-1)),s[153]||(s[153]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[154]||(s[154]=e('
julia
fno(; setup, kmax, c, σ, ψ, rng, kwargs...)
Create FNO closure model. Return a tuple (closure, θ) where θ are the initial parameters and closure(V, θ) predicts the commutator error.
source
',3))]),s[178]||(s[178]=t("h2",{id:"Data-generation",tabindex:"-1"},[a("Data generation "),t("a",{class:"header-anchor",href:"#Data-generation","aria-label":'Permalink to "Data generation {#Data-generation}"'},"​")],-1)),t("details",L1,[t("summary",null,[s[155]||(s[155]=t("a",{id:"NeuralClosure.create_io_arrays-Tuple{Any, Any}",href:"#NeuralClosure.create_io_arrays-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"NeuralClosure.create_io_arrays")],-1)),s[156]||(s[156]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[161]||(s[161]=e('
julia
create_io_arrays(data, setup) -> NamedTuple
',1)),t("p",null,[s[159]||(s[159]=a("Create ")),t("mjx-container",V1,[(o(),n("svg",N1,s[157]||(s[157]=[e('',1)]))),s[158]||(s[158]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mo",{stretchy:"false"},"("),t("mrow",{"data-mjx-texclass":"ORD"},[t("mover",null,[t("mi",null,"u"),t("mo",{stretchy:"false"},"¯")])]),t("mo",null,","),t("mi",null,"c"),t("mo",{stretchy:"false"},")")])],-1))]),s[160]||(s[160]=a(" pairs for a-priori training."))]),s[162]||(s[162]=t("p",null,[t("a",{href:"https://github.com/agdestein/IncompressibleNavierStokes.jl/blob/03acafa3566742b1f37d722d2207f5dc21d031c0/lib/NeuralClosure/src/data_generation.jl#L228",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",Z1,[t("summary",null,[s[163]||(s[163]=t("a",{id:"NeuralClosure.create_les_data-Tuple{}",href:"#NeuralClosure.create_les_data-Tuple{}"},[t("span",{class:"jlbinding"},"NeuralClosure.create_les_data")],-1)),s[164]||(s[164]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[165]||(s[165]=e(`
julia
create_les_data(
+)
Create CNN closure model. Return a tuple (closure, θ) where θ are the initial parameters and closure(u, θ) predicts the commutator error.
source
`,3))]),t("details",f1,[t("summary",null,[s[131]||(s[131]=t("a",{id:"NeuralClosure.GroupConv2D",href:"#NeuralClosure.GroupConv2D"},[t("span",{class:"jlbinding"},"NeuralClosure.GroupConv2D")],-1)),s[132]||(s[132]=a()),i(l,{type:"info",class:"jlObjectType jlType",text:"Type"})]),s[133]||(s[133]=e('
julia
struct GroupConv2D{C} <: LuxCore.AbstractLuxLayer
Group-equivariant convolutional layer – with respect to the p4 group. The layer is equivariant to rotations and translations of the input vector field.
The kwargs are passed to the Conv layer.
The layer has three variants:
  • If islifting then it lifts a vector input (u1, u2) into a rotation-state vector (v1, v2, v3, v4).
  • If isprojecting, it projects a rotation-state vector (u1, u2, u3, v4) into a vector (v1, v2).
  • Otherwise, it cyclically transforms the rotation-state vector (u1, u2, u3, u4) into a new rotation-state vector (v1, v2, v3, v4).
Fields
  • islifting
  • isprojecting
  • cin
  • cout
  • conv
source
',8))]),t("details",E1,[t("summary",null,[s[134]||(s[134]=t("a",{id:"NeuralClosure.gcnn-Tuple{}",href:"#NeuralClosure.gcnn-Tuple{}"},[t("span",{class:"jlbinding"},"NeuralClosure.gcnn")],-1)),s[135]||(s[135]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[136]||(s[136]=e('
julia
gcnn(; setup, radii, channels, activations, use_bias, rng)
Create CNN closure model. Return a tuple (closure, θ) where θ are the initial parameters and closure(u, θ) predicts the commutator error.
source
',3))]),t("details",w1,[t("summary",null,[s[137]||(s[137]=t("a",{id:"NeuralClosure.rot2",href:"#NeuralClosure.rot2"},[t("span",{class:"jlbinding"},"NeuralClosure.rot2")],-1)),s[138]||(s[138]=a()),i(l,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[139]||(s[139]=e('
julia
rot2(u, r)
Rotate the field u by 90 degrees counter-clockwise r - 1 times.
source
',3))]),t("details",v1,[t("summary",null,[s[140]||(s[140]=t("a",{id:"NeuralClosure.rot2-Union{Tuple{T}, Tuple{Tuple{T, T}, Any}} where T",href:"#NeuralClosure.rot2-Union{Tuple{T}, Tuple{Tuple{T, T}, Any}} where T"},[t("span",{class:"jlbinding"},"NeuralClosure.rot2")],-1)),s[141]||(s[141]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[142]||(s[142]=e('
julia
rot2(u::Tuple{T, T}, r) -> Union{Nothing, Tuple{Any, Any}}
Rotate vector fields [ux;;; uy]
source
',3))]),t("details",H1,[t("summary",null,[s[143]||(s[143]=t("a",{id:"NeuralClosure.rot2stag-Tuple{Any, Any}",href:"#NeuralClosure.rot2stag-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"NeuralClosure.rot2stag")],-1)),s[144]||(s[144]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[145]||(s[145]=e('
julia
rot2stag(u, g) -> Any
Rotate staggered grid velocity field. See also rot2.
source
',3))]),t("details",j1,[t("summary",null,[s[146]||(s[146]=t("a",{id:"NeuralClosure.vecrot2-Tuple{Any, Any}",href:"#NeuralClosure.vecrot2-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"NeuralClosure.vecrot2")],-1)),s[147]||(s[147]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[148]||(s[148]=e('
julia
vecrot2(u, r) -> Any
Rotate vector fields [ux;;; uy]
source
',3))]),t("details",C1,[t("summary",null,[s[149]||(s[149]=t("a",{id:"NeuralClosure.FourierLayer",href:"#NeuralClosure.FourierLayer"},[t("span",{class:"jlbinding"},"NeuralClosure.FourierLayer")],-1)),s[150]||(s[150]=a()),i(l,{type:"info",class:"jlObjectType jlType",text:"Type"})]),s[151]||(s[151]=e('
julia
struct FourierLayer{D, A, F} <: LuxCore.AbstractLuxLayer
Fourier layer operating on uniformly discretized functions.
Some important sizes:
  • dimension: Spatial dimension, e.g. Dimension(2) or Dimension(3).
  • (nx..., cin, nsample): Input size
  • (nx..., cout, nsample): Output size
  • nx = fill(n, dimension()): Number of points in each spatial dimension
  • n ≥ kmax: Same number of points in each spatial dimension, must be larger than cut-off wavenumber
  • kmax: Cut-off wavenumber
  • nsample: Number of input samples (treated independently)
Fields
  • dimension
  • kmax
  • cin
  • cout
  • σ
  • init_weight
source
',7))]),t("details",M1,[t("summary",null,[s[152]||(s[152]=t("a",{id:"NeuralClosure.fno-Tuple{}",href:"#NeuralClosure.fno-Tuple{}"},[t("span",{class:"jlbinding"},"NeuralClosure.fno")],-1)),s[153]||(s[153]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[154]||(s[154]=e('
julia
fno(; setup, kmax, c, σ, ψ, rng, kwargs...)
Create FNO closure model. Return a tuple (closure, θ) where θ are the initial parameters and closure(V, θ) predicts the commutator error.
source
',3))]),s[178]||(s[178]=t("h2",{id:"Data-generation",tabindex:"-1"},[a("Data generation "),t("a",{class:"header-anchor",href:"#Data-generation","aria-label":'Permalink to "Data generation {#Data-generation}"'},"​")],-1)),t("details",L1,[t("summary",null,[s[155]||(s[155]=t("a",{id:"NeuralClosure.create_io_arrays-Tuple{Any, Any}",href:"#NeuralClosure.create_io_arrays-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"NeuralClosure.create_io_arrays")],-1)),s[156]||(s[156]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[161]||(s[161]=e('
julia
create_io_arrays(data, setup) -> NamedTuple
',1)),t("p",null,[s[159]||(s[159]=a("Create ")),t("mjx-container",V1,[(o(),n("svg",N1,s[157]||(s[157]=[e('',1)]))),s[158]||(s[158]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mo",{stretchy:"false"},"("),t("mrow",{"data-mjx-texclass":"ORD"},[t("mover",null,[t("mi",null,"u"),t("mo",{stretchy:"false"},"¯")])]),t("mo",null,","),t("mi",null,"c"),t("mo",{stretchy:"false"},")")])],-1))]),s[160]||(s[160]=a(" pairs for a-priori training."))]),s[162]||(s[162]=t("p",null,[t("a",{href:"https://github.com/agdestein/IncompressibleNavierStokes.jl/blob/a63d20e3a410f14ba6cc612a27e89f8974019d9f/lib/NeuralClosure/src/data_generation.jl#L228",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",Z1,[t("summary",null,[s[163]||(s[163]=t("a",{id:"NeuralClosure.create_les_data-Tuple{}",href:"#NeuralClosure.create_les_data-Tuple{}"},[t("span",{class:"jlbinding"},"NeuralClosure.create_les_data")],-1)),s[164]||(s[164]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[165]||(s[165]=e(`
julia
create_les_data(
 ;
     D,
     Re,
@@ -98,7 +98,7 @@ import{_ as r,c as n,a5 as e,j as t,a,G as i,B as d,o}from"./chunks/framework.Dr
     rng,
     filenames,
     kwargs...
-)
Create filtered DNS data.
source
`,3))]),t("details",D1,[t("summary",null,[s[166]||(s[166]=t("a",{id:"NeuralClosure.filtersaver-NTuple{6, Any}",href:"#NeuralClosure.filtersaver-NTuple{6, Any}"},[t("span",{class:"jlbinding"},"NeuralClosure.filtersaver")],-1)),s[167]||(s[167]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[168]||(s[168]=e(`
julia
filtersaver(
+)
Create filtered DNS data.
source
`,3))]),t("details",D1,[t("summary",null,[s[166]||(s[166]=t("a",{id:"NeuralClosure.filtersaver-NTuple{6, Any}",href:"#NeuralClosure.filtersaver-NTuple{6, Any}"},[t("span",{class:"jlbinding"},"NeuralClosure.filtersaver")],-1)),s[167]||(s[167]=a()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[168]||(s[168]=e(`
julia
filtersaver(
     dns,
     les,
     filters,
@@ -109,4 +109,4 @@ import{_ as r,c as n,a5 as e,j as t,a,G as i,B as d,o}from"./chunks/framework.Dr
     filenames,
     F,
     p
-)
Save filtered DNS data.
source
`,3))])])}const G1=r(p,[["render",F1]]);export{z1 as __pageData,G1 as default};
+)
Save filtered DNS data.
source
`,3))])])}const G1=r(p,[["render",F1]]);export{z1 as __pageData,G1 as default};
diff --git a/dev/assets/manual_differentiability.md.DHl4CtMi.js b/dev/assets/manual_differentiability.md.D_UkzJTY.js
similarity index 89%
rename from dev/assets/manual_differentiability.md.DHl4CtMi.js
rename to dev/assets/manual_differentiability.md.D_UkzJTY.js
index 9564a486a..ef7eecea5 100644
--- a/dev/assets/manual_differentiability.md.DHl4CtMi.js
+++ b/dev/assets/manual_differentiability.md.D_UkzJTY.js
@@ -88,6 +88,6 @@ import{_ as p,c as n,a5 as e,j as i,a,o as t}from"./chunks/framework.Dr1kDDE8.js
   (dft-vrank>=1-x51/1
     (dft-ct-dit/10
       (dftw-direct-10/6 "t3fv_10_avx2_128")
-      (dft-directbuf/14-10-x10 "n1fv_10_avx2")))), [6.945445343448e-310 6.9454449186883e-310 … 5.0e-324 5.0e-324; 6.9454449184535e-310 6.94544491869066e-310 … 6.94544489383323e-310 6.9454448939123e-310; … ; 6.94544491868354e-310 6.9454449003201e-310 … 5.0e-324 5.0e-324; 6.9454449186859e-310 6.9454449003225e-310 … 6.9454448939107e-310 6.94544489398975e-310], ComplexF64[6.94557386934597e-310 + 6.94557386934597e-310im 0.0 + 0.0im … 0.0 + 0.0im 0.0 + 0.0im; 6.94557386934597e-310 + 6.94557386934597e-310im 0.0 + 0.0im … 0.0 + 0.0im 0.0 + 0.0im; … ; 0.0 + 0.0im 0.0 + 0.0im … 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im … 0.0 + 0.0im 0.0 + 0.0im], Core.Box(([0.0, 0.003946543143456876, 0.01577059737104434, 0.035425498542622634, 0.06283367774273778, 0.09788696740969285, 0.1404470282234972, 0.19034589506796099, 0.24738663991227286, 0.3113441489959698  …  3.68865585100403, 3.752613360087727, 3.809654104932039, 3.859552971776503, 3.9021130325903073, 3.9371663222572626, 3.9645745014573777, 3.984229402628956, 3.996053456856544, 4.0], [0.0, 0.003946543143456876, 0.01577059737104434, 0.035425498542622634, 0.06283367774273778, 0.09788696740969285, 0.1404470282234972, 0.19034589506796099, 0.24738663991227286, 0.3113441489959698  …  0.3819660112501049, 0.3113441489959696, 0.24738663991227255, 0.19034589506796068, 0.14044702822349744, 0.09788696740969302, 0.06283367774273789, 0.0354254985426227, 0.015770597371044366, 0.003946543143456883])), 2, CartesianIndices((2:101, 2:101)))])
Now, we call the autodiff function from Enzyme:
julia
Enzyme.autodiff(Enzyme.Reverse, f!, Duplicated(dudt,ddudt), Duplicated(u,du), Const(params_ref), Const(t))
((nothing, nothing, nothing, nothing),)
Since we have passed a Duplicated object, the gradient of u is stored in du.
Finally, we can also compare its value with the one obtained by Zygote differentiating the out-of-place (non-mutating) version of the right-hand side:
julia
f = create_right_hand_side(setup, psolver)
+      (dft-directbuf/14-10-x10 "n1fv_10_avx2")))), [3.86215878e-315 0.0 … 6.548379943e-315 3.89362187e-316; 6.547993426e-315 0.0 … 6.548380615e-315 3.89362187e-316; … ; 4.245092511e-314 3.22322206e-315 … 6.34907335e-315 0.0; 0.0 6.91711403163036e-310 … 2.9e-322 6.34907303e-315], ComplexF64[1.89668687e-315 + 2.20435554e-315im 5.0e-324 + 6.91697151413794e-310im … 1.191723655e-315 + 4.9e-322im 3.0355e-320 + 6.37e-322im; 1.89668687e-315 + 2.20435554e-315im 5.0e-324 + 6.91697151413952e-310im … 3.9e-322 + 5.0e-324im 5.3e-322 + 3.05534496e-315im; … ; 5.0e-324 + 6.9169715141348e-310im 5.0e-324 + 6.9169715142154e-310im … 5.24e-322 + 2.3716543e-316im 6.4e-322 + 3.955624e-316im; 5.0e-324 + 6.91697151413636e-310im 5.0e-324 + 6.916971514217e-310im … 6.4e-322 + 5.3e-322im 7.9e-322 + 6.47e-322im], Core.Box(([0.0, 0.003946543143456876, 0.01577059737104434, 0.035425498542622634, 0.06283367774273778, 0.09788696740969285, 0.1404470282234972, 0.19034589506796099, 0.24738663991227286, 0.3113441489959698  …  3.68865585100403, 3.752613360087727, 3.809654104932039, 3.859552971776503, 3.9021130325903073, 3.9371663222572626, 3.9645745014573777, 3.984229402628956, 3.996053456856544, 4.0], [0.0, 0.003946543143456876, 0.01577059737104434, 0.035425498542622634, 0.06283367774273778, 0.09788696740969285, 0.1404470282234972, 0.19034589506796099, 0.24738663991227286, 0.3113441489959698  …  0.3819660112501049, 0.3113441489959696, 0.24738663991227255, 0.19034589506796068, 0.14044702822349744, 0.09788696740969302, 0.06283367774273789, 0.0354254985426227, 0.015770597371044366, 0.003946543143456883])), 2, CartesianIndices((2:101, 2:101)))])
Now, we call the autodiff function from Enzyme:
julia
Enzyme.autodiff(Enzyme.Reverse, f!, Duplicated(dudt,ddudt), Duplicated(u,du), Const(params_ref), Const(t))
((nothing, nothing, nothing, nothing),)
Since we have passed a Duplicated object, the gradient of u is stored in du.
Finally, we can also compare its value with the one obtained by Zygote differentiating the out-of-place (non-mutating) version of the right-hand side:
julia
f = create_right_hand_side(setup, psolver)
 _, zpull = Zygote.pullback(f, u, nothing, 0.0);
 @assert zpull(dudt)[1] == du
`,9))])}const m=p(l,[["render",r]]);export{u as __pageData,m as default};
diff --git a/dev/assets/manual_differentiability.md.DHl4CtMi.lean.js b/dev/assets/manual_differentiability.md.D_UkzJTY.lean.js
similarity index 89%
rename from dev/assets/manual_differentiability.md.DHl4CtMi.lean.js
rename to dev/assets/manual_differentiability.md.D_UkzJTY.lean.js
index 9564a486a..ef7eecea5 100644
--- a/dev/assets/manual_differentiability.md.DHl4CtMi.lean.js
+++ b/dev/assets/manual_differentiability.md.D_UkzJTY.lean.js
@@ -88,6 +88,6 @@ import{_ as p,c as n,a5 as e,j as i,a,o as t}from"./chunks/framework.Dr1kDDE8.js
   (dft-vrank>=1-x51/1
     (dft-ct-dit/10
       (dftw-direct-10/6 "t3fv_10_avx2_128")
-      (dft-directbuf/14-10-x10 "n1fv_10_avx2")))), [6.945445343448e-310 6.9454449186883e-310 … 5.0e-324 5.0e-324; 6.9454449184535e-310 6.94544491869066e-310 … 6.94544489383323e-310 6.9454448939123e-310; … ; 6.94544491868354e-310 6.9454449003201e-310 … 5.0e-324 5.0e-324; 6.9454449186859e-310 6.9454449003225e-310 … 6.9454448939107e-310 6.94544489398975e-310], ComplexF64[6.94557386934597e-310 + 6.94557386934597e-310im 0.0 + 0.0im … 0.0 + 0.0im 0.0 + 0.0im; 6.94557386934597e-310 + 6.94557386934597e-310im 0.0 + 0.0im … 0.0 + 0.0im 0.0 + 0.0im; … ; 0.0 + 0.0im 0.0 + 0.0im … 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im … 0.0 + 0.0im 0.0 + 0.0im], Core.Box(([0.0, 0.003946543143456876, 0.01577059737104434, 0.035425498542622634, 0.06283367774273778, 0.09788696740969285, 0.1404470282234972, 0.19034589506796099, 0.24738663991227286, 0.3113441489959698  …  3.68865585100403, 3.752613360087727, 3.809654104932039, 3.859552971776503, 3.9021130325903073, 3.9371663222572626, 3.9645745014573777, 3.984229402628956, 3.996053456856544, 4.0], [0.0, 0.003946543143456876, 0.01577059737104434, 0.035425498542622634, 0.06283367774273778, 0.09788696740969285, 0.1404470282234972, 0.19034589506796099, 0.24738663991227286, 0.3113441489959698  …  0.3819660112501049, 0.3113441489959696, 0.24738663991227255, 0.19034589506796068, 0.14044702822349744, 0.09788696740969302, 0.06283367774273789, 0.0354254985426227, 0.015770597371044366, 0.003946543143456883])), 2, CartesianIndices((2:101, 2:101)))])
Now, we call the autodiff function from Enzyme:
julia
Enzyme.autodiff(Enzyme.Reverse, f!, Duplicated(dudt,ddudt), Duplicated(u,du), Const(params_ref), Const(t))
((nothing, nothing, nothing, nothing),)
Since we have passed a Duplicated object, the gradient of u is stored in du.
Finally, we can also compare its value with the one obtained by Zygote differentiating the out-of-place (non-mutating) version of the right-hand side:
julia
f = create_right_hand_side(setup, psolver)
+      (dft-directbuf/14-10-x10 "n1fv_10_avx2")))), [3.86215878e-315 0.0 … 6.548379943e-315 3.89362187e-316; 6.547993426e-315 0.0 … 6.548380615e-315 3.89362187e-316; … ; 4.245092511e-314 3.22322206e-315 … 6.34907335e-315 0.0; 0.0 6.91711403163036e-310 … 2.9e-322 6.34907303e-315], ComplexF64[1.89668687e-315 + 2.20435554e-315im 5.0e-324 + 6.91697151413794e-310im … 1.191723655e-315 + 4.9e-322im 3.0355e-320 + 6.37e-322im; 1.89668687e-315 + 2.20435554e-315im 5.0e-324 + 6.91697151413952e-310im … 3.9e-322 + 5.0e-324im 5.3e-322 + 3.05534496e-315im; … ; 5.0e-324 + 6.9169715141348e-310im 5.0e-324 + 6.9169715142154e-310im … 5.24e-322 + 2.3716543e-316im 6.4e-322 + 3.955624e-316im; 5.0e-324 + 6.91697151413636e-310im 5.0e-324 + 6.916971514217e-310im … 6.4e-322 + 5.3e-322im 7.9e-322 + 6.47e-322im], Core.Box(([0.0, 0.003946543143456876, 0.01577059737104434, 0.035425498542622634, 0.06283367774273778, 0.09788696740969285, 0.1404470282234972, 0.19034589506796099, 0.24738663991227286, 0.3113441489959698  …  3.68865585100403, 3.752613360087727, 3.809654104932039, 3.859552971776503, 3.9021130325903073, 3.9371663222572626, 3.9645745014573777, 3.984229402628956, 3.996053456856544, 4.0], [0.0, 0.003946543143456876, 0.01577059737104434, 0.035425498542622634, 0.06283367774273778, 0.09788696740969285, 0.1404470282234972, 0.19034589506796099, 0.24738663991227286, 0.3113441489959698  …  0.3819660112501049, 0.3113441489959696, 0.24738663991227255, 0.19034589506796068, 0.14044702822349744, 0.09788696740969302, 0.06283367774273789, 0.0354254985426227, 0.015770597371044366, 0.003946543143456883])), 2, CartesianIndices((2:101, 2:101)))])
Now, we call the autodiff function from Enzyme:
julia
Enzyme.autodiff(Enzyme.Reverse, f!, Duplicated(dudt,ddudt), Duplicated(u,du), Const(params_ref), Const(t))
((nothing, nothing, nothing, nothing),)
Since we have passed a Duplicated object, the gradient of u is stored in du.
Finally, we can also compare its value with the one obtained by Zygote differentiating the out-of-place (non-mutating) version of the right-hand side:
julia
f = create_right_hand_side(setup, psolver)
 _, zpull = Zygote.pullback(f, u, nothing, 0.0);
 @assert zpull(dudt)[1] == du
`,9))])}const m=p(l,[["render",r]]);export{u as __pageData,m as default};
diff --git a/dev/assets/manual_matrices.md.CgitaAGh.js b/dev/assets/manual_matrices.md.WgRK4vkk.js
similarity index 98%
rename from dev/assets/manual_matrices.md.CgitaAGh.js
rename to dev/assets/manual_matrices.md.WgRK4vkk.js
index d7942052e..03c3c9c8c 100644
--- a/dev/assets/manual_matrices.md.CgitaAGh.js
+++ b/dev/assets/manual_matrices.md.WgRK4vkk.js
@@ -76,7 +76,7 @@ import{_ as r,c as l,a5 as n,j as a,a as i,G as t,B as o,o as p}from"./chunks/fr
 D = diffusion_mat(setup)
 d_kernel = diffusion(apply_bc_u(u, 0.0, setup), setup; use_viscosity = false)
 d_matrix = reshape(D * B * u[:], size(u))
-maximum(abs, d_matrix - d_kernel)
9.094947017729282e-13
`,5)),a("p",null,[s[15]||(s[15]=i("Matrices only work on flattened fields ")),s[16]||(s[16]=a("code",null,"u[:]",-1)),s[17]||(s[17]=i(", while the kernels work on ")),a("mjx-container",m,[(p(),l("svg",u,s[11]||(s[11]=[n('',1)]))),s[12]||(s[12]=a("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("mo",{stretchy:"false"},"("),a("mi",null,"D"),a("mo",null,"+"),a("mn",null,"1"),a("mo",{stretchy:"false"},")")])],-1))]),s[18]||(s[18]=i("-array-shaped fields for a dimension ")),a("mjx-container",E,[(p(),l("svg",y,s[13]||(s[13]=[n('',1)]))),s[14]||(s[14]=a("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("mi",null,"D"),a("mo",null,"∈"),a("mo",{fence:"false",stretchy:"false"},"{"),a("mn",null,"2"),a("mo",null,","),a("mn",null,"3"),a("mo",{fence:"false",stretchy:"false"},"}")])],-1))]),s[19]||(s[19]=i("."))]),s[55]||(s[55]=n(`
Boundary conditions and matrices ​
Matrices can only be used to represent boundary conditions that depend linearly on the input, such as periodic or Neumann boundary conditions. Non-zero Dirichlet boundary conditions need to be accounted for separately. Consider the following inflow-setup:
julia
setup = Setup(;
+maximum(abs, d_matrix - d_kernel)
4.547473508864641e-13
`,5)),a("p",null,[s[15]||(s[15]=i("Matrices only work on flattened fields ")),s[16]||(s[16]=a("code",null,"u[:]",-1)),s[17]||(s[17]=i(", while the kernels work on ")),a("mjx-container",m,[(p(),l("svg",u,s[11]||(s[11]=[n('',1)]))),s[12]||(s[12]=a("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("mo",{stretchy:"false"},"("),a("mi",null,"D"),a("mo",null,"+"),a("mn",null,"1"),a("mo",{stretchy:"false"},")")])],-1))]),s[18]||(s[18]=i("-array-shaped fields for a dimension ")),a("mjx-container",E,[(p(),l("svg",y,s[13]||(s[13]=[n('',1)]))),s[14]||(s[14]=a("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("mi",null,"D"),a("mo",null,"∈"),a("mo",{fence:"false",stretchy:"false"},"{"),a("mn",null,"2"),a("mo",null,","),a("mn",null,"3"),a("mo",{fence:"false",stretchy:"false"},"}")])],-1))]),s[19]||(s[19]=i("."))]),s[55]||(s[55]=n(`
Boundary conditions and matrices ​
Matrices can only be used to represent boundary conditions that depend linearly on the input, such as periodic or Neumann boundary conditions. Non-zero Dirichlet boundary conditions need to be accounted for separately. Consider the following inflow-setup:
julia
setup = Setup(;
     x = (ax, ax),
     boundary_conditions = (
         (DirichletBC((10.0, 0.0)), PressureBC()),
@@ -108,12 +108,12 @@ import{_ as r,c as l,a5 as n,j as a,a as i,G as t,B as o,o as p}from"./chunks/fr
  0.0     0.0     0.0     0.0     0.0        0.0     0.0     0.0     0.0  0.0
  0.0     0.0     0.0     0.0     0.0  …     0.0     0.0     0.0     0.0  0.0
  0.0     0.0     0.0     0.0     0.0        0.0     0.0     0.0     0.0  0.0
- 0.0     0.0     0.0     0.0     0.0        0.0     0.0     0.0     0.0  0.0
By adding yM, we get the equality:
julia
maximum(abs, (div_matrix + yM) - div_kernel)
2.842170943040401e-14
API ​
`,14)),a("details",b,[a("summary",null,[s[20]||(s[20]=a("a",{id:"IncompressibleNavierStokes.bc_p_mat",href:"#IncompressibleNavierStokes.bc_p_mat"},[a("span",{class:"jlbinding"},"IncompressibleNavierStokes.bc_p_mat")],-1)),s[21]||(s[21]=i()),t(e,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[22]||(s[22]=a("p",null,[i("Matrix for applying boundary conditions to pressure fields "),a("code",null,"p"),i(".")],-1)),s[23]||(s[23]=a("p",null,[a("a",{href:"https://github.com/agdestein/IncompressibleNavierStokes.jl/blob/03acafa3566742b1f37d722d2207f5dc21d031c0/src/matrices.jl#L61",target:"_blank",rel:"noreferrer"},"source")],-1))]),a("details",T,[a("summary",null,[s[24]||(s[24]=a("a",{id:"IncompressibleNavierStokes.bc_temp_mat",href:"#IncompressibleNavierStokes.bc_temp_mat"},[a("span",{class:"jlbinding"},"IncompressibleNavierStokes.bc_temp_mat")],-1)),s[25]||(s[25]=i()),t(e,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[26]||(s[26]=a("p",null,[i("Matrix for applying boundary conditions to temperature fields "),a("code",null,"temp"),i(".")],-1)),s[27]||(s[27]=a("p",null,[a("a",{href:"https://github.com/agdestein/IncompressibleNavierStokes.jl/blob/03acafa3566742b1f37d722d2207f5dc21d031c0/src/matrices.jl#L64",target:"_blank",rel:"noreferrer"},"source")],-1))]),a("details",Q,[a("summary",null,[s[28]||(s[28]=a("a",{id:"IncompressibleNavierStokes.bc_u_mat",href:"#IncompressibleNavierStokes.bc_u_mat"},[a("span",{class:"jlbinding"},"IncompressibleNavierStokes.bc_u_mat")],-1)),s[29]||(s[29]=i()),t(e,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[30]||(s[30]=a("p",null,[i("Create matrix for applying boundary conditions to velocity fields "),a("code",null,"u"),i(". This matrix only applies the boundary conditions depending on "),a("code",null,"u"),i(" itself (e.g. "),a("a",{href:"/IncompressibleNavierStokes.jl/dev/manual/setup#IncompressibleNavierStokes.PeriodicBC"},[a("code",null,"PeriodicBC")]),i("). It does not apply constant boundary conditions (e.g. non-zero "),a("a",{href:"/IncompressibleNavierStokes.jl/dev/manual/setup#IncompressibleNavierStokes.DirichletBC"},[a("code",null,"DirichletBC")]),i(").")],-1)),s[31]||(s[31]=a("p",null,[a("a",{href:"https://github.com/agdestein/IncompressibleNavierStokes.jl/blob/03acafa3566742b1f37d722d2207f5dc21d031c0/src/matrices.jl#L54",target:"_blank",rel:"noreferrer"},"source")],-1))]),a("details",v,[a("summary",null,[s[32]||(s[32]=a("a",{id:"IncompressibleNavierStokes.diffusion_mat-Tuple{Any}",href:"#IncompressibleNavierStokes.diffusion_mat-Tuple{Any}"},[a("span",{class:"jlbinding"},"IncompressibleNavierStokes.diffusion_mat")],-1)),s[33]||(s[33]=i()),t(e,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[34]||(s[34]=n(`
julia
diffusion_mat(
+ 0.0     0.0     0.0     0.0     0.0        0.0     0.0     0.0     0.0  0.0
By adding yM, we get the equality:
julia
maximum(abs, (div_matrix + yM) - div_kernel)
2.842170943040401e-14
API ​
`,14)),a("details",b,[a("summary",null,[s[20]||(s[20]=a("a",{id:"IncompressibleNavierStokes.bc_p_mat",href:"#IncompressibleNavierStokes.bc_p_mat"},[a("span",{class:"jlbinding"},"IncompressibleNavierStokes.bc_p_mat")],-1)),s[21]||(s[21]=i()),t(e,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[22]||(s[22]=a("p",null,[i("Matrix for applying boundary conditions to pressure fields "),a("code",null,"p"),i(".")],-1)),s[23]||(s[23]=a("p",null,[a("a",{href:"https://github.com/agdestein/IncompressibleNavierStokes.jl/blob/a63d20e3a410f14ba6cc612a27e89f8974019d9f/src/matrices.jl#L61",target:"_blank",rel:"noreferrer"},"source")],-1))]),a("details",T,[a("summary",null,[s[24]||(s[24]=a("a",{id:"IncompressibleNavierStokes.bc_temp_mat",href:"#IncompressibleNavierStokes.bc_temp_mat"},[a("span",{class:"jlbinding"},"IncompressibleNavierStokes.bc_temp_mat")],-1)),s[25]||(s[25]=i()),t(e,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[26]||(s[26]=a("p",null,[i("Matrix for applying boundary conditions to temperature fields "),a("code",null,"temp"),i(".")],-1)),s[27]||(s[27]=a("p",null,[a("a",{href:"https://github.com/agdestein/IncompressibleNavierStokes.jl/blob/a63d20e3a410f14ba6cc612a27e89f8974019d9f/src/matrices.jl#L64",target:"_blank",rel:"noreferrer"},"source")],-1))]),a("details",Q,[a("summary",null,[s[28]||(s[28]=a("a",{id:"IncompressibleNavierStokes.bc_u_mat",href:"#IncompressibleNavierStokes.bc_u_mat"},[a("span",{class:"jlbinding"},"IncompressibleNavierStokes.bc_u_mat")],-1)),s[29]||(s[29]=i()),t(e,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[30]||(s[30]=a("p",null,[i("Create matrix for applying boundary conditions to velocity fields "),a("code",null,"u"),i(". This matrix only applies the boundary conditions depending on "),a("code",null,"u"),i(" itself (e.g. "),a("a",{href:"/IncompressibleNavierStokes.jl/dev/manual/setup#IncompressibleNavierStokes.PeriodicBC"},[a("code",null,"PeriodicBC")]),i("). It does not apply constant boundary conditions (e.g. non-zero "),a("a",{href:"/IncompressibleNavierStokes.jl/dev/manual/setup#IncompressibleNavierStokes.DirichletBC"},[a("code",null,"DirichletBC")]),i(").")],-1)),s[31]||(s[31]=a("p",null,[a("a",{href:"https://github.com/agdestein/IncompressibleNavierStokes.jl/blob/a63d20e3a410f14ba6cc612a27e89f8974019d9f/src/matrices.jl#L54",target:"_blank",rel:"noreferrer"},"source")],-1))]),a("details",v,[a("summary",null,[s[32]||(s[32]=a("a",{id:"IncompressibleNavierStokes.diffusion_mat-Tuple{Any}",href:"#IncompressibleNavierStokes.diffusion_mat-Tuple{Any}"},[a("span",{class:"jlbinding"},"IncompressibleNavierStokes.diffusion_mat")],-1)),s[33]||(s[33]=i()),t(e,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[34]||(s[34]=n(`
julia
diffusion_mat(
     setup
-) -> SparseArrays.SparseMatrixCSC{Tv, Int64} where Tv
Diffusion matrix.
source
`,3))]),a("details",f,[a("summary",null,[s[35]||(s[35]=a("a",{id:"IncompressibleNavierStokes.divergence_mat-Tuple{Any}",href:"#IncompressibleNavierStokes.divergence_mat-Tuple{Any}"},[a("span",{class:"jlbinding"},"IncompressibleNavierStokes.divergence_mat")],-1)),s[36]||(s[36]=i()),t(e,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[37]||(s[37]=n(`
julia
divergence_mat(
+) -> SparseArrays.SparseMatrixCSC{Tv, Int64} where Tv
Diffusion matrix.
source
`,3))]),a("details",f,[a("summary",null,[s[35]||(s[35]=a("a",{id:"IncompressibleNavierStokes.divergence_mat-Tuple{Any}",href:"#IncompressibleNavierStokes.divergence_mat-Tuple{Any}"},[a("span",{class:"jlbinding"},"IncompressibleNavierStokes.divergence_mat")],-1)),s[36]||(s[36]=i()),t(e,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[37]||(s[37]=n(`
julia
divergence_mat(
     setup
-) -> SparseArrays.SparseMatrixCSC{Tv, Int64} where Tv
Divergence matrix.
source
`,3))]),a("details",C,[a("summary",null,[s[38]||(s[38]=a("a",{id:"IncompressibleNavierStokes.laplacian_mat-Tuple{Any}",href:"#IncompressibleNavierStokes.laplacian_mat-Tuple{Any}"},[a("span",{class:"jlbinding"},"IncompressibleNavierStokes.laplacian_mat")],-1)),s[39]||(s[39]=i()),t(e,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[40]||(s[40]=n('
julia
laplacian_mat(setup) -> Any
Get matrix for the Laplacian operator (for the pressure-Poisson equation). This matrix takes scalar field inputs restricted to the actual degrees of freedom.
source
',3))]),a("details",x,[a("summary",null,[s[41]||(s[41]=a("a",{id:"IncompressibleNavierStokes.pad_scalarfield_mat-Tuple{Any}",href:"#IncompressibleNavierStokes.pad_scalarfield_mat-Tuple{Any}"},[a("span",{class:"jlbinding"},"IncompressibleNavierStokes.pad_scalarfield_mat")],-1)),s[42]||(s[42]=i()),t(e,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[43]||(s[43]=n('
julia
pad_scalarfield_mat(setup) -> Any
Create matrix for padding inner scalar field with boundary volumes. This can be useful for algorithms that require vectors with degrees of freedom only, and not the ghost volumes. To go back, simply transpose the matrix.
See also: pad_vectorfield_mat.
source
',4))]),a("details",F,[a("summary",null,[s[44]||(s[44]=a("a",{id:"IncompressibleNavierStokes.pad_vectorfield_mat-Tuple{Any}",href:"#IncompressibleNavierStokes.pad_vectorfield_mat-Tuple{Any}"},[a("span",{class:"jlbinding"},"IncompressibleNavierStokes.pad_vectorfield_mat")],-1)),s[45]||(s[45]=i()),t(e,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[46]||(s[46]=n(`
julia
pad_vectorfield_mat(
+) -> SparseArrays.SparseMatrixCSC{Tv, Int64} where Tv
Divergence matrix.
source
`,3))]),a("details",C,[a("summary",null,[s[38]||(s[38]=a("a",{id:"IncompressibleNavierStokes.laplacian_mat-Tuple{Any}",href:"#IncompressibleNavierStokes.laplacian_mat-Tuple{Any}"},[a("span",{class:"jlbinding"},"IncompressibleNavierStokes.laplacian_mat")],-1)),s[39]||(s[39]=i()),t(e,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[40]||(s[40]=n('
julia
laplacian_mat(setup) -> Any
Get matrix for the Laplacian operator (for the pressure-Poisson equation). This matrix takes scalar field inputs restricted to the actual degrees of freedom.
source
',3))]),a("details",x,[a("summary",null,[s[41]||(s[41]=a("a",{id:"IncompressibleNavierStokes.pad_scalarfield_mat-Tuple{Any}",href:"#IncompressibleNavierStokes.pad_scalarfield_mat-Tuple{Any}"},[a("span",{class:"jlbinding"},"IncompressibleNavierStokes.pad_scalarfield_mat")],-1)),s[42]||(s[42]=i()),t(e,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[43]||(s[43]=n('
julia
pad_scalarfield_mat(setup) -> Any
Create matrix for padding inner scalar field with boundary volumes. This can be useful for algorithms that require vectors with degrees of freedom only, and not the ghost volumes. To go back, simply transpose the matrix.
See also: pad_vectorfield_mat.
source
',4))]),a("details",F,[a("summary",null,[s[44]||(s[44]=a("a",{id:"IncompressibleNavierStokes.pad_vectorfield_mat-Tuple{Any}",href:"#IncompressibleNavierStokes.pad_vectorfield_mat-Tuple{Any}"},[a("span",{class:"jlbinding"},"IncompressibleNavierStokes.pad_vectorfield_mat")],-1)),s[45]||(s[45]=i()),t(e,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[46]||(s[46]=n(`
julia
pad_vectorfield_mat(
     setup
-) -> SparseArrays.SparseMatrixCSC{Tv, Int64} where Tv
Create matrix for padding inner vector field with boundary volumes, similar to pad_scalarfield_mat.
source
`,3))]),a("details",j,[a("summary",null,[s[47]||(s[47]=a("a",{id:"IncompressibleNavierStokes.pressuregradient_mat-Tuple{Any}",href:"#IncompressibleNavierStokes.pressuregradient_mat-Tuple{Any}"},[a("span",{class:"jlbinding"},"IncompressibleNavierStokes.pressuregradient_mat")],-1)),s[48]||(s[48]=i()),t(e,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[49]||(s[49]=n(`
julia
pressuregradient_mat(
+) -> SparseArrays.SparseMatrixCSC{Tv, Int64} where Tv
Create matrix for padding inner vector field with boundary volumes, similar to pad_scalarfield_mat.
source
`,3))]),a("details",j,[a("summary",null,[s[47]||(s[47]=a("a",{id:"IncompressibleNavierStokes.pressuregradient_mat-Tuple{Any}",href:"#IncompressibleNavierStokes.pressuregradient_mat-Tuple{Any}"},[a("span",{class:"jlbinding"},"IncompressibleNavierStokes.pressuregradient_mat")],-1)),s[48]||(s[48]=i()),t(e,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[49]||(s[49]=n(`
julia
pressuregradient_mat(
     setup
-) -> SparseArrays.SparseMatrixCSC{Tv, Int64} where Tv
Pressure gradient matrix.
source
`,3))]),a("details",w,[a("summary",null,[s[50]||(s[50]=a("a",{id:"IncompressibleNavierStokes.volume_mat-Tuple{Any}",href:"#IncompressibleNavierStokes.volume_mat-Tuple{Any}"},[a("span",{class:"jlbinding"},"IncompressibleNavierStokes.volume_mat")],-1)),s[51]||(s[51]=i()),t(e,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[52]||(s[52]=n('
julia
volume_mat(setup) -> Any
Volume-size matrix.
source
',3))])])}const _=r(h,[["render",S]]);export{L as __pageData,_ as default};
+) -> SparseArrays.SparseMatrixCSC{Tv, Int64} where Tv
Pressure gradient matrix.
source
`,3))]),a("details",w,[a("summary",null,[s[50]||(s[50]=a("a",{id:"IncompressibleNavierStokes.volume_mat-Tuple{Any}",href:"#IncompressibleNavierStokes.volume_mat-Tuple{Any}"},[a("span",{class:"jlbinding"},"IncompressibleNavierStokes.volume_mat")],-1)),s[51]||(s[51]=i()),t(e,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[52]||(s[52]=n('
julia
volume_mat(setup) -> Any
Volume-size matrix.
source
',3))])])}const _=r(h,[["render",S]]);export{L as __pageData,_ as default};
diff --git a/dev/assets/manual_matrices.md.CgitaAGh.lean.js b/dev/assets/manual_matrices.md.WgRK4vkk.lean.js
similarity index 98%
rename from dev/assets/manual_matrices.md.CgitaAGh.lean.js
rename to dev/assets/manual_matrices.md.WgRK4vkk.lean.js
index d7942052e..03c3c9c8c 100644
--- a/dev/assets/manual_matrices.md.CgitaAGh.lean.js
+++ b/dev/assets/manual_matrices.md.WgRK4vkk.lean.js
@@ -76,7 +76,7 @@ import{_ as r,c as l,a5 as n,j as a,a as i,G as t,B as o,o as p}from"./chunks/fr
 D = diffusion_mat(setup)
 d_kernel = diffusion(apply_bc_u(u, 0.0, setup), setup; use_viscosity = false)
 d_matrix = reshape(D * B * u[:], size(u))
-maximum(abs, d_matrix - d_kernel)
9.094947017729282e-13
`,5)),a("p",null,[s[15]||(s[15]=i("Matrices only work on flattened fields ")),s[16]||(s[16]=a("code",null,"u[:]",-1)),s[17]||(s[17]=i(", while the kernels work on ")),a("mjx-container",m,[(p(),l("svg",u,s[11]||(s[11]=[n('',1)]))),s[12]||(s[12]=a("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("mo",{stretchy:"false"},"("),a("mi",null,"D"),a("mo",null,"+"),a("mn",null,"1"),a("mo",{stretchy:"false"},")")])],-1))]),s[18]||(s[18]=i("-array-shaped fields for a dimension ")),a("mjx-container",E,[(p(),l("svg",y,s[13]||(s[13]=[n('',1)]))),s[14]||(s[14]=a("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("mi",null,"D"),a("mo",null,"∈"),a("mo",{fence:"false",stretchy:"false"},"{"),a("mn",null,"2"),a("mo",null,","),a("mn",null,"3"),a("mo",{fence:"false",stretchy:"false"},"}")])],-1))]),s[19]||(s[19]=i("."))]),s[55]||(s[55]=n(`
Boundary conditions and matrices ​
Matrices can only be used to represent boundary conditions that depend linearly on the input, such as periodic or Neumann boundary conditions. Non-zero Dirichlet boundary conditions need to be accounted for separately. Consider the following inflow-setup:
julia
setup = Setup(;
+maximum(abs, d_matrix - d_kernel)
4.547473508864641e-13
`,5)),a("p",null,[s[15]||(s[15]=i("Matrices only work on flattened fields ")),s[16]||(s[16]=a("code",null,"u[:]",-1)),s[17]||(s[17]=i(", while the kernels work on ")),a("mjx-container",m,[(p(),l("svg",u,s[11]||(s[11]=[n('',1)]))),s[12]||(s[12]=a("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("mo",{stretchy:"false"},"("),a("mi",null,"D"),a("mo",null,"+"),a("mn",null,"1"),a("mo",{stretchy:"false"},")")])],-1))]),s[18]||(s[18]=i("-array-shaped fields for a dimension ")),a("mjx-container",E,[(p(),l("svg",y,s[13]||(s[13]=[n('',1)]))),s[14]||(s[14]=a("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("mi",null,"D"),a("mo",null,"∈"),a("mo",{fence:"false",stretchy:"false"},"{"),a("mn",null,"2"),a("mo",null,","),a("mn",null,"3"),a("mo",{fence:"false",stretchy:"false"},"}")])],-1))]),s[19]||(s[19]=i("."))]),s[55]||(s[55]=n(`
Boundary conditions and matrices ​
Matrices can only be used to represent boundary conditions that depend linearly on the input, such as periodic or Neumann boundary conditions. Non-zero Dirichlet boundary conditions need to be accounted for separately. Consider the following inflow-setup:
julia
setup = Setup(;
     x = (ax, ax),
     boundary_conditions = (
         (DirichletBC((10.0, 0.0)), PressureBC()),
@@ -108,12 +108,12 @@ import{_ as r,c as l,a5 as n,j as a,a as i,G as t,B as o,o as p}from"./chunks/fr
  0.0     0.0     0.0     0.0     0.0        0.0     0.0     0.0     0.0  0.0
  0.0     0.0     0.0     0.0     0.0  …     0.0     0.0     0.0     0.0  0.0
  0.0     0.0     0.0     0.0     0.0        0.0     0.0     0.0     0.0  0.0
- 0.0     0.0     0.0     0.0     0.0        0.0     0.0     0.0     0.0  0.0
By adding yM, we get the equality:
julia
maximum(abs, (div_matrix + yM) - div_kernel)
2.842170943040401e-14
API ​
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julia
diffusion_mat(
+ 0.0     0.0     0.0     0.0     0.0        0.0     0.0     0.0     0.0  0.0
By adding yM, we get the equality:
julia
maximum(abs, (div_matrix + yM) - div_kernel)
2.842170943040401e-14
API ​
`,14)),a("details",b,[a("summary",null,[s[20]||(s[20]=a("a",{id:"IncompressibleNavierStokes.bc_p_mat",href:"#IncompressibleNavierStokes.bc_p_mat"},[a("span",{class:"jlbinding"},"IncompressibleNavierStokes.bc_p_mat")],-1)),s[21]||(s[21]=i()),t(e,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[22]||(s[22]=a("p",null,[i("Matrix for applying boundary conditions to pressure fields "),a("code",null,"p"),i(".")],-1)),s[23]||(s[23]=a("p",null,[a("a",{href:"https://github.com/agdestein/IncompressibleNavierStokes.jl/blob/a63d20e3a410f14ba6cc612a27e89f8974019d9f/src/matrices.jl#L61",target:"_blank",rel:"noreferrer"},"source")],-1))]),a("details",T,[a("summary",null,[s[24]||(s[24]=a("a",{id:"IncompressibleNavierStokes.bc_temp_mat",href:"#IncompressibleNavierStokes.bc_temp_mat"},[a("span",{class:"jlbinding"},"IncompressibleNavierStokes.bc_temp_mat")],-1)),s[25]||(s[25]=i()),t(e,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[26]||(s[26]=a("p",null,[i("Matrix for applying boundary conditions to temperature fields "),a("code",null,"temp"),i(".")],-1)),s[27]||(s[27]=a("p",null,[a("a",{href:"https://github.com/agdestein/IncompressibleNavierStokes.jl/blob/a63d20e3a410f14ba6cc612a27e89f8974019d9f/src/matrices.jl#L64",target:"_blank",rel:"noreferrer"},"source")],-1))]),a("details",Q,[a("summary",null,[s[28]||(s[28]=a("a",{id:"IncompressibleNavierStokes.bc_u_mat",href:"#IncompressibleNavierStokes.bc_u_mat"},[a("span",{class:"jlbinding"},"IncompressibleNavierStokes.bc_u_mat")],-1)),s[29]||(s[29]=i()),t(e,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[30]||(s[30]=a("p",null,[i("Create matrix for applying boundary conditions to velocity fields "),a("code",null,"u"),i(". This matrix only applies the boundary conditions depending on "),a("code",null,"u"),i(" itself (e.g. "),a("a",{href:"/IncompressibleNavierStokes.jl/dev/manual/setup#IncompressibleNavierStokes.PeriodicBC"},[a("code",null,"PeriodicBC")]),i("). It does not apply constant boundary conditions (e.g. non-zero "),a("a",{href:"/IncompressibleNavierStokes.jl/dev/manual/setup#IncompressibleNavierStokes.DirichletBC"},[a("code",null,"DirichletBC")]),i(").")],-1)),s[31]||(s[31]=a("p",null,[a("a",{href:"https://github.com/agdestein/IncompressibleNavierStokes.jl/blob/a63d20e3a410f14ba6cc612a27e89f8974019d9f/src/matrices.jl#L54",target:"_blank",rel:"noreferrer"},"source")],-1))]),a("details",v,[a("summary",null,[s[32]||(s[32]=a("a",{id:"IncompressibleNavierStokes.diffusion_mat-Tuple{Any}",href:"#IncompressibleNavierStokes.diffusion_mat-Tuple{Any}"},[a("span",{class:"jlbinding"},"IncompressibleNavierStokes.diffusion_mat")],-1)),s[33]||(s[33]=i()),t(e,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[34]||(s[34]=n(`
julia
diffusion_mat(
     setup
-) -> SparseArrays.SparseMatrixCSC{Tv, Int64} where Tv
Diffusion matrix.
source
`,3))]),a("details",f,[a("summary",null,[s[35]||(s[35]=a("a",{id:"IncompressibleNavierStokes.divergence_mat-Tuple{Any}",href:"#IncompressibleNavierStokes.divergence_mat-Tuple{Any}"},[a("span",{class:"jlbinding"},"IncompressibleNavierStokes.divergence_mat")],-1)),s[36]||(s[36]=i()),t(e,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[37]||(s[37]=n(`
julia
divergence_mat(
+) -> SparseArrays.SparseMatrixCSC{Tv, Int64} where Tv
Diffusion matrix.
source
`,3))]),a("details",f,[a("summary",null,[s[35]||(s[35]=a("a",{id:"IncompressibleNavierStokes.divergence_mat-Tuple{Any}",href:"#IncompressibleNavierStokes.divergence_mat-Tuple{Any}"},[a("span",{class:"jlbinding"},"IncompressibleNavierStokes.divergence_mat")],-1)),s[36]||(s[36]=i()),t(e,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[37]||(s[37]=n(`
julia
divergence_mat(
     setup
-) -> SparseArrays.SparseMatrixCSC{Tv, Int64} where Tv
Divergence matrix.
source
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julia
laplacian_mat(setup) -> Any
Get matrix for the Laplacian operator (for the pressure-Poisson equation). This matrix takes scalar field inputs restricted to the actual degrees of freedom.
source
',3))]),a("details",x,[a("summary",null,[s[41]||(s[41]=a("a",{id:"IncompressibleNavierStokes.pad_scalarfield_mat-Tuple{Any}",href:"#IncompressibleNavierStokes.pad_scalarfield_mat-Tuple{Any}"},[a("span",{class:"jlbinding"},"IncompressibleNavierStokes.pad_scalarfield_mat")],-1)),s[42]||(s[42]=i()),t(e,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[43]||(s[43]=n('
julia
pad_scalarfield_mat(setup) -> Any
Create matrix for padding inner scalar field with boundary volumes. This can be useful for algorithms that require vectors with degrees of freedom only, and not the ghost volumes. To go back, simply transpose the matrix.
See also: pad_vectorfield_mat.
source
',4))]),a("details",F,[a("summary",null,[s[44]||(s[44]=a("a",{id:"IncompressibleNavierStokes.pad_vectorfield_mat-Tuple{Any}",href:"#IncompressibleNavierStokes.pad_vectorfield_mat-Tuple{Any}"},[a("span",{class:"jlbinding"},"IncompressibleNavierStokes.pad_vectorfield_mat")],-1)),s[45]||(s[45]=i()),t(e,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[46]||(s[46]=n(`
julia
pad_vectorfield_mat(
+) -> SparseArrays.SparseMatrixCSC{Tv, Int64} where Tv
Divergence matrix.
source
`,3))]),a("details",C,[a("summary",null,[s[38]||(s[38]=a("a",{id:"IncompressibleNavierStokes.laplacian_mat-Tuple{Any}",href:"#IncompressibleNavierStokes.laplacian_mat-Tuple{Any}"},[a("span",{class:"jlbinding"},"IncompressibleNavierStokes.laplacian_mat")],-1)),s[39]||(s[39]=i()),t(e,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[40]||(s[40]=n('
julia
laplacian_mat(setup) -> Any
Get matrix for the Laplacian operator (for the pressure-Poisson equation). This matrix takes scalar field inputs restricted to the actual degrees of freedom.
source
',3))]),a("details",x,[a("summary",null,[s[41]||(s[41]=a("a",{id:"IncompressibleNavierStokes.pad_scalarfield_mat-Tuple{Any}",href:"#IncompressibleNavierStokes.pad_scalarfield_mat-Tuple{Any}"},[a("span",{class:"jlbinding"},"IncompressibleNavierStokes.pad_scalarfield_mat")],-1)),s[42]||(s[42]=i()),t(e,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[43]||(s[43]=n('
julia
pad_scalarfield_mat(setup) -> Any
Create matrix for padding inner scalar field with boundary volumes. This can be useful for algorithms that require vectors with degrees of freedom only, and not the ghost volumes. To go back, simply transpose the matrix.
See also: pad_vectorfield_mat.
source
',4))]),a("details",F,[a("summary",null,[s[44]||(s[44]=a("a",{id:"IncompressibleNavierStokes.pad_vectorfield_mat-Tuple{Any}",href:"#IncompressibleNavierStokes.pad_vectorfield_mat-Tuple{Any}"},[a("span",{class:"jlbinding"},"IncompressibleNavierStokes.pad_vectorfield_mat")],-1)),s[45]||(s[45]=i()),t(e,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[46]||(s[46]=n(`
julia
pad_vectorfield_mat(
     setup
-) -> SparseArrays.SparseMatrixCSC{Tv, Int64} where Tv
Create matrix for padding inner vector field with boundary volumes, similar to pad_scalarfield_mat.
source
`,3))]),a("details",j,[a("summary",null,[s[47]||(s[47]=a("a",{id:"IncompressibleNavierStokes.pressuregradient_mat-Tuple{Any}",href:"#IncompressibleNavierStokes.pressuregradient_mat-Tuple{Any}"},[a("span",{class:"jlbinding"},"IncompressibleNavierStokes.pressuregradient_mat")],-1)),s[48]||(s[48]=i()),t(e,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[49]||(s[49]=n(`
julia
pressuregradient_mat(
+) -> SparseArrays.SparseMatrixCSC{Tv, Int64} where Tv
Create matrix for padding inner vector field with boundary volumes, similar to pad_scalarfield_mat.
source
`,3))]),a("details",j,[a("summary",null,[s[47]||(s[47]=a("a",{id:"IncompressibleNavierStokes.pressuregradient_mat-Tuple{Any}",href:"#IncompressibleNavierStokes.pressuregradient_mat-Tuple{Any}"},[a("span",{class:"jlbinding"},"IncompressibleNavierStokes.pressuregradient_mat")],-1)),s[48]||(s[48]=i()),t(e,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[49]||(s[49]=n(`
julia
pressuregradient_mat(
     setup
-) -> SparseArrays.SparseMatrixCSC{Tv, Int64} where Tv
Pressure gradient matrix.
source
`,3))]),a("details",w,[a("summary",null,[s[50]||(s[50]=a("a",{id:"IncompressibleNavierStokes.volume_mat-Tuple{Any}",href:"#IncompressibleNavierStokes.volume_mat-Tuple{Any}"},[a("span",{class:"jlbinding"},"IncompressibleNavierStokes.volume_mat")],-1)),s[51]||(s[51]=i()),t(e,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[52]||(s[52]=n('
julia
volume_mat(setup) -> Any
Volume-size matrix.
source
',3))])])}const _=r(h,[["render",S]]);export{L as __pageData,_ as default};
+) -> SparseArrays.SparseMatrixCSC{Tv, Int64} where Tv
Pressure gradient matrix.
source
`,3))]),a("details",w,[a("summary",null,[s[50]||(s[50]=a("a",{id:"IncompressibleNavierStokes.volume_mat-Tuple{Any}",href:"#IncompressibleNavierStokes.volume_mat-Tuple{Any}"},[a("span",{class:"jlbinding"},"IncompressibleNavierStokes.volume_mat")],-1)),s[51]||(s[51]=i()),t(e,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[52]||(s[52]=n('
julia
volume_mat(setup) -> Any
Volume-size matrix.
source
',3))])])}const _=r(h,[["render",S]]);export{L as __pageData,_ as default};
diff --git a/dev/assets/manual_operators.md.d5WWXEo-.lean.js b/dev/assets/manual_operators.md.DDNX2oaK.js
similarity index 97%
rename from dev/assets/manual_operators.md.d5WWXEo-.lean.js
rename to dev/assets/manual_operators.md.DDNX2oaK.js
index eee6825e6..353a61433 100644
--- a/dev/assets/manual_operators.md.d5WWXEo-.lean.js
+++ b/dev/assets/manual_operators.md.DDNX2oaK.js
@@ -1,6 +1,6 @@
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Operators ​
All discrete operators are built using KernelAbstractions.jl and Cartesian indices, similar to WaterLily.jl. This allows for dimension- and backend-agnostic code. See this blog post for how to write kernels. IncompressibleNavierStokes previously relied on assembling sparse operators to perform the same operations. While being very efficient and also compatible with CUDA (CUSPARSE), storing these matrices in memory is expensive for large 3D problems.
',2)),t("details",d,[t("summary",null,[s[0]||(s[0]=t("a",{id:"IncompressibleNavierStokes.Offset",href:"#IncompressibleNavierStokes.Offset"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.Offset")],-1)),s[1]||(s[1]=e()),i(l,{type:"info",class:"jlObjectType jlType",text:"Type"})]),s[2]||(s[2]=a('
julia
struct Offset{D}
Cartesian index unit vector in D = 2 or D = 3 dimensions. Calling Offset(D)(α) returns a Cartesian index with 1 in the dimension α and zeros elsewhere.
See https://b-fg.github.io/2023/05/07/waterlily-on-gpu.html for writing kernel loops using Cartesian indices.
Fields
source
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julia
Dfield!(d, G, p, setup; ϵ) -> Any
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julia
Dfield(p, setup; kwargs...) -> Any
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julia
Qfield!(Q, u, setup) -> Any
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julia
Qfield(u, setup) -> Any
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julia
applybodyforce!(F, u, t, setup) -> Any
Compute body force (in-place version). Add the result to F.
source
',3))]),t("details",M,[t("summary",null,[s[48]||(s[48]=t("a",{id:"IncompressibleNavierStokes.applybodyforce-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.applybodyforce-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.applybodyforce")],-1)),s[49]||(s[49]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[50]||(s[50]=a('
julia
applybodyforce(u, t, setup) -> Any
Compute body force (differentiable version).
source
',3))]),t("details",A,[t("summary",null,[s[51]||(s[51]=t("a",{id:"IncompressibleNavierStokes.applypressure!-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.applypressure!-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.applypressure!")],-1)),s[52]||(s[52]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[53]||(s[53]=a('
julia
applypressure!(u, p, setup) -> Any
Subtract pressure gradient (in-place version).
source
',3))]),t("details",C,[t("summary",null,[s[54]||(s[54]=t("a",{id:"IncompressibleNavierStokes.applypressure-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.applypressure-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.applypressure")],-1)),s[55]||(s[55]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[56]||(s[56]=a('
julia
applypressure(u, p, setup) -> Any
Subtract pressure gradient (differentiable version).
source
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julia
avg(ϕ, Δ, I, α) -> Any
Average scalar field ϕ in the α-direction.
source
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julia
convection!(F, u, setup) -> Any
Compute convective term (in-place version). Add the result to F.
source
',3))]),t("details",S,[t("summary",null,[s[63]||(s[63]=t("a",{id:"IncompressibleNavierStokes.convection-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.convection-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.convection")],-1)),s[64]||(s[64]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[65]||(s[65]=a('
julia
convection(u, setup) -> Any
Compute convective term (differentiable version).
source
',3))]),t("details",N,[t("summary",null,[s[66]||(s[66]=t("a",{id:"IncompressibleNavierStokes.convection_diffusion_temp!-NTuple{4, Any}",href:"#IncompressibleNavierStokes.convection_diffusion_temp!-NTuple{4, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.convection_diffusion_temp!")],-1)),s[67]||(s[67]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[68]||(s[68]=a('
julia
convection_diffusion_temp!(c, u, temp, setup) -> Any
Compute convection-diffusion term for the temperature equation. (in-place version). Add result to c.
source
',3))]),t("details",I,[t("summary",null,[s[69]||(s[69]=t("a",{id:"IncompressibleNavierStokes.convection_diffusion_temp-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.convection_diffusion_temp-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.convection_diffusion_temp")],-1)),s[70]||(s[70]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[71]||(s[71]=a('
julia
convection_diffusion_temp(u, temp, setup) -> Any
Compute convection-diffusion term for the temperature equation. (differentiable version).
source
',3))]),t("details",D,[t("summary",null,[s[72]||(s[72]=t("a",{id:"IncompressibleNavierStokes.convectiondiffusion!-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.convectiondiffusion!-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.convectiondiffusion!")],-1)),s[73]||(s[73]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[74]||(s[74]=a('
julia
convectiondiffusion!(F, u, setup) -> Any
Compute convective and diffusive terms (in-place version). Add the result to F.
source
',3))]),t("details",F,[t("summary",null,[s[75]||(s[75]=t("a",{id:"IncompressibleNavierStokes.diffusion!-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.diffusion!-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.diffusion!")],-1)),s[76]||(s[76]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[77]||(s[77]=a('
julia
diffusion!(F, u, setup; use_viscosity) -> Any
Compute diffusive term (in-place version). Add the result to F.
Keyword arguments:
  • with_viscosity = true: Include viscosity in the operator.
source
',5))]),t("details",O,[t("summary",null,[s[78]||(s[78]=t("a",{id:"IncompressibleNavierStokes.diffusion-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.diffusion-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.diffusion")],-1)),s[79]||(s[79]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[80]||(s[80]=a('
julia
diffusion(u, setup; kwargs...) -> Any
Compute diffusive term (differentiable version).
source
',3))]),t("details",B,[t("summary",null,[s[81]||(s[81]=t("a",{id:"IncompressibleNavierStokes.dissipation!-NTuple{4, Any}",href:"#IncompressibleNavierStokes.dissipation!-NTuple{4, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.dissipation!")],-1)),s[82]||(s[82]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[83]||(s[83]=a('
julia
dissipation!(diss, diff, u, setup) -> Any
Compute dissipation term for the temperature equation (in-place version). Add result to diss.
source
',3))]),t("details",R,[t("summary",null,[s[84]||(s[84]=t("a",{id:"IncompressibleNavierStokes.dissipation-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.dissipation-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.dissipation")],-1)),s[85]||(s[85]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[86]||(s[86]=a('
julia
dissipation(u, setup) -> Any
Compute dissipation term for the temperature equation (differentiable version).
source
',3))]),t("details",G,[t("summary",null,[s[87]||(s[87]=t("a",{id:"IncompressibleNavierStokes.dissipation_from_strain!-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.dissipation_from_strain!-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.dissipation_from_strain!")],-1)),s[88]||(s[88]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[89]||(s[89]=a('
julia
dissipation_from_strain!(ϵ, u, setup) -> Any
Compute dissipation term from strain-rate tensor (in-place version).
source
',3))]),t("details",J,[t("summary",null,[s[90]||(s[90]=t("a",{id:"IncompressibleNavierStokes.dissipation_from_strain-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.dissipation_from_strain-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.dissipation_from_strain")],-1)),s[91]||(s[91]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[96]||(s[96]=a('
julia
dissipation_from_strain(u, setup) -> Any
',1)),t("p",null,[s[94]||(s[94]=e("Compute dissipation term ")),t("mjx-container",z,[(o(),n("svg",X,s[92]||(s[92]=[a('',1)]))),s[93]||(s[93]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mn",null,"2"),t("mi",null,"ν"),t("mo",{fence:"false",stretchy:"false"},"⟨"),t("msub",null,[t("mi",null,"S"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",null,"i"),t("mi",null,"j")])]),t("msub",null,[t("mi",null,"S"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",null,"i"),t("mi",null,"j")])]),t("mo",{fence:"false",stretchy:"false"},"⟩")])],-1))]),s[95]||(s[95]=e(" from strain-rate tensor (differentiable version)."))]),s[97]||(s[97]=t("p",null,[t("a",{href:"https://github.com/agdestein/IncompressibleNavierStokes.jl/blob/03acafa3566742b1f37d722d2207f5dc21d031c0/src/operators.jl#L810",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",q,[t("summary",null,[s[98]||(s[98]=t("a",{id:"IncompressibleNavierStokes.divergence!-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.divergence!-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.divergence!")],-1)),s[99]||(s[99]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[100]||(s[100]=a('
julia
divergence!(div, u, setup) -> Any
Compute divergence of velocity field (in-place version).
source
',3))]),t("details",P,[t("summary",null,[s[101]||(s[101]=t("a",{id:"IncompressibleNavierStokes.divergence-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.divergence-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.divergence")],-1)),s[102]||(s[102]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[103]||(s[103]=a('
julia
divergence(u, setup) -> Any
Compute divergence of velocity field (differentiable version).
source
',3))]),t("details",K,[t("summary",null,[s[104]||(s[104]=t("a",{id:"IncompressibleNavierStokes.divoftensor!-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.divoftensor!-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.divoftensor!")],-1)),s[105]||(s[105]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[106]||(s[106]=a('
julia
divoftensor!(s, σ, setup) -> Any
Compute divergence of a tensor with all components in the pressure points (in-place version). The stress tensors should be precomputed and stored in σ.
source
',3))]),t("details",U,[t("summary",null,[s[107]||(s[107]=t("a",{id:"IncompressibleNavierStokes.eig2field!-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.eig2field!-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.eig2field!")],-1)),s[108]||(s[108]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[113]||(s[113]=a('
julia
eig2field!(λ, u, setup) -> Any
',1)),t("p",null,[s[111]||(s[111]=e("Compute the second eigenvalue of ")),t("mjx-container",$,[(o(),n("svg",W,s[109]||(s[109]=[a('',1)]))),s[110]||(s[110]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msup",null,[t("mi",null,"S"),t("mn",null,"2")]),t("mo",null,"+"),t("msup",null,[t("mi",null,"R"),t("mn",null,"2")])])],-1))]),s[112]||(s[112]=e(" (in-place version)."))]),s[114]||(s[114]=t("p",null,[t("a",{href:"https://github.com/agdestein/IncompressibleNavierStokes.jl/blob/03acafa3566742b1f37d722d2207f5dc21d031c0/src/operators.jl#L1470",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",Y,[t("summary",null,[s[115]||(s[115]=t("a",{id:"IncompressibleNavierStokes.eig2field-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.eig2field-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.eig2field")],-1)),s[116]||(s[116]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[123]||(s[123]=a('
julia
eig2field(u, setup) -> Any
',1)),t("p",null,[s[119]||(s[119]=e("Compute the second eigenvalue of ")),t("mjx-container",_,[(o(),n("svg",t1,s[117]||(s[117]=[a('',1)]))),s[118]||(s[118]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msup",null,[t("mi",null,"S"),t("mn",null,"2")]),t("mo",null,"+"),t("msup",null,[t("mi",null,"R"),t("mn",null,"2")])])],-1))]),s[120]||(s[120]=e(", as proposed by Jeong and Hussain [")),s[121]||(s[121]=t("a",{href:"/IncompressibleNavierStokes.jl/dev/references#Jeong1995"},"6",-1)),s[122]||(s[122]=e("]."))]),s[124]||(s[124]=t("p",null,"Differentiable version.",-1)),s[125]||(s[125]=t("p",null,[t("a",{href:"https://github.com/agdestein/IncompressibleNavierStokes.jl/blob/03acafa3566742b1f37d722d2207f5dc21d031c0/src/operators.jl#L1462",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",s1,[t("summary",null,[s[126]||(s[126]=t("a",{id:"IncompressibleNavierStokes.get_scale_numbers-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.get_scale_numbers-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.get_scale_numbers")],-1)),s[127]||(s[127]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[149]||(s[149]=a(`
julia
get_scale_numbers(
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Operators ​
All discrete operators are built using KernelAbstractions.jl and Cartesian indices, similar to WaterLily.jl. This allows for dimension- and backend-agnostic code. See this blog post for how to write kernels. IncompressibleNavierStokes previously relied on assembling sparse operators to perform the same operations. While being very efficient and also compatible with CUDA (CUSPARSE), storing these matrices in memory is expensive for large 3D problems.
',2)),t("details",d,[t("summary",null,[s[0]||(s[0]=t("a",{id:"IncompressibleNavierStokes.Offset",href:"#IncompressibleNavierStokes.Offset"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.Offset")],-1)),s[1]||(s[1]=e()),i(l,{type:"info",class:"jlObjectType jlType",text:"Type"})]),s[2]||(s[2]=a('
julia
struct Offset{D}
Cartesian index unit vector in D = 2 or D = 3 dimensions. Calling Offset(D)(α) returns a Cartesian index with 1 in the dimension α and zeros elsewhere.
See https://b-fg.github.io/2023/05/07/waterlily-on-gpu.html for writing kernel loops using Cartesian indices.
Fields
source
',5))]),t("details",p,[t("summary",null,[s[3]||(s[3]=t("a",{id:"IncompressibleNavierStokes.Dfield!-NTuple{4, Any}",href:"#IncompressibleNavierStokes.Dfield!-NTuple{4, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.Dfield!")],-1)),s[4]||(s[4]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[9]||(s[9]=a('
julia
Dfield!(d, G, p, setup; ϵ) -> Any
',1)),t("p",null,[s[7]||(s[7]=e("Compute the ")),t("mjx-container",m,[(o(),n("svg",h,s[5]||(s[5]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D437",d:"M287 628Q287 635 230 637Q207 637 200 638T193 647Q193 655 197 667T204 682Q206 683 403 683Q570 682 590 682T630 676Q702 659 752 597T803 431Q803 275 696 151T444 3L430 1L236 0H125H72Q48 0 41 2T33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM703 469Q703 507 692 537T666 584T629 613T590 629T555 636Q553 636 541 636T512 636T479 637H436Q392 637 386 627Q384 623 313 339T242 52Q242 48 253 48T330 47Q335 47 349 47T373 46Q499 46 581 128Q617 164 640 212T683 339T703 469Z",style:{"stroke-width":"3"}})])])],-1)]))),s[6]||(s[6]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"D")])],-1))]),s[8]||(s[8]=e("-field (in-place version)."))]),s[10]||(s[10]=t("p",null,[t("a",{href:"https://github.com/agdestein/IncompressibleNavierStokes.jl/blob/a63d20e3a410f14ba6cc612a27e89f8974019d9f/src/operators.jl#L1389",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",g,[t("summary",null,[s[11]||(s[11]=t("a",{id:"IncompressibleNavierStokes.Dfield-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.Dfield-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.Dfield")],-1)),s[12]||(s[12]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[21]||(s[21]=a('
julia
Dfield(p, setup; kwargs...) -> Any
',1)),t("p",null,[s[15]||(s[15]=e("Compute the ")),t("mjx-container",u,[(o(),n("svg",k,s[13]||(s[13]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D437",d:"M287 628Q287 635 230 637Q207 637 200 638T193 647Q193 655 197 667T204 682Q206 683 403 683Q570 682 590 682T630 676Q702 659 752 597T803 431Q803 275 696 151T444 3L430 1L236 0H125H72Q48 0 41 2T33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM703 469Q703 507 692 537T666 584T629 613T590 629T555 636Q553 636 541 636T512 636T479 637H436Q392 637 386 627Q384 623 313 339T242 52Q242 48 253 48T330 47Q335 47 349 47T373 46Q499 46 581 128Q617 164 640 212T683 339T703 469Z",style:{"stroke-width":"3"}})])])],-1)]))),s[14]||(s[14]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"D")])],-1))]),s[16]||(s[16]=e("-field [")),s[17]||(s[17]=t("a",{href:"/IncompressibleNavierStokes.jl/dev/references#LiJiajia2019"},"5",-1)),s[18]||(s[18]=e("] given by"))]),t("mjx-container",c,[(o(),n("svg",y,s[19]||(s[19]=[a('',1)]))),s[20]||(s[20]=t("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[t("mi",null,"D"),t("mo",null,"="),t("mfrac",null,[t("mrow",null,[t("mn",null,"2"),t("mo",{"data-mjx-texclass":"ORD",stretchy:"false"},"|"),t("mi",{mathvariant:"normal"},"∇"),t("mi",null,"p"),t("mo",{"data-mjx-texclass":"ORD",stretchy:"false"},"|")]),t("mrow",null,[t("msup",null,[t("mi",{mathvariant:"normal"},"∇"),t("mn",null,"2")]),t("mi",null,"p")])]),t("mo",null,".")])],-1))]),s[22]||(s[22]=t("p",null,"Differentiable version.",-1)),s[23]||(s[23]=t("p",null,[t("a",{href:"https://github.com/agdestein/IncompressibleNavierStokes.jl/blob/a63d20e3a410f14ba6cc612a27e89f8974019d9f/src/operators.jl#L1374",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",b,[t("summary",null,[s[24]||(s[24]=t("a",{id:"IncompressibleNavierStokes.Qfield!-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.Qfield!-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.Qfield!")],-1)),s[25]||(s[25]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[30]||(s[30]=a('
julia
Qfield!(Q, u, setup) -> Any
',1)),t("p",null,[s[28]||(s[28]=e("Compute the ")),t("mjx-container",f,[(o(),n("svg",v,s[26]||(s[26]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D444",d:"M399 -80Q399 -47 400 -30T402 -11V-7L387 -11Q341 -22 303 -22Q208 -22 138 35T51 201Q50 209 50 244Q50 346 98 438T227 601Q351 704 476 704Q514 704 524 703Q621 689 680 617T740 435Q740 255 592 107Q529 47 461 16L444 8V3Q444 2 449 -24T470 -66T516 -82Q551 -82 583 -60T625 -3Q631 11 638 11Q647 11 649 2Q649 -6 639 -34T611 -100T557 -165T481 -194Q399 -194 399 -87V-80ZM636 468Q636 523 621 564T580 625T530 655T477 665Q429 665 379 640Q277 591 215 464T153 216Q153 110 207 59Q231 38 236 38V46Q236 86 269 120T347 155Q372 155 390 144T417 114T429 82T435 55L448 64Q512 108 557 185T619 334T636 468ZM314 18Q362 18 404 39L403 49Q399 104 366 115Q354 117 347 117Q344 117 341 117T337 118Q317 118 296 98T274 52Q274 18 314 18Z",style:{"stroke-width":"3"}})])])],-1)]))),s[27]||(s[27]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"Q")])],-1))]),s[29]||(s[29]=e("-field (in-place version)."))]),s[31]||(s[31]=t("p",null,[t("a",{href:"https://github.com/agdestein/IncompressibleNavierStokes.jl/blob/a63d20e3a410f14ba6cc612a27e89f8974019d9f/src/operators.jl#L1440",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",x,[t("summary",null,[s[32]||(s[32]=t("a",{id:"IncompressibleNavierStokes.Qfield-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.Qfield-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.Qfield")],-1)),s[33]||(s[33]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[42]||(s[42]=a('
julia
Qfield(u, setup) -> Any
',1)),t("p",null,[s[36]||(s[36]=e("Compute ")),t("mjx-container",w,[(o(),n("svg",j,s[34]||(s[34]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D444",d:"M399 -80Q399 -47 400 -30T402 -11V-7L387 -11Q341 -22 303 -22Q208 -22 138 35T51 201Q50 209 50 244Q50 346 98 438T227 601Q351 704 476 704Q514 704 524 703Q621 689 680 617T740 435Q740 255 592 107Q529 47 461 16L444 8V3Q444 2 449 -24T470 -66T516 -82Q551 -82 583 -60T625 -3Q631 11 638 11Q647 11 649 2Q649 -6 639 -34T611 -100T557 -165T481 -194Q399 -194 399 -87V-80ZM636 468Q636 523 621 564T580 625T530 655T477 665Q429 665 379 640Q277 591 215 464T153 216Q153 110 207 59Q231 38 236 38V46Q236 86 269 120T347 155Q372 155 390 144T417 114T429 82T435 55L448 64Q512 108 557 185T619 334T636 468ZM314 18Q362 18 404 39L403 49Q399 104 366 115Q354 117 347 117Q344 117 341 117T337 118Q317 118 296 98T274 52Q274 18 314 18Z",style:{"stroke-width":"3"}})])])],-1)]))),s[35]||(s[35]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"Q")])],-1))]),s[37]||(s[37]=e("-field [")),s[38]||(s[38]=t("a",{href:"/IncompressibleNavierStokes.jl/dev/references#Jeong1995"},"6",-1)),s[39]||(s[39]=e("] given by"))]),t("mjx-container",H,[(o(),n("svg",L,s[40]||(s[40]=[a('',1)]))),s[41]||(s[41]=t("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[t("mi",null,"Q"),t("mo",null,"="),t("mo",null,"−"),t("mfrac",null,[t("mn",null,"1"),t("mn",null,"2")]),t("munder",null,[t("mo",{"data-mjx-texclass":"OP"},"∑"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",null,"α"),t("mo",null,","),t("mi",null,"β")])]),t("mfrac",null,[t("mrow",null,[t("mi",null,"∂"),t("msup",null,[t("mi",null,"u"),t("mi",null,"α")])]),t("mrow",null,[t("mi",null,"∂"),t("msup",null,[t("mi",null,"x"),t("mi",null,"β")])])]),t("mfrac",null,[t("mrow",null,[t("mi",null,"∂"),t("msup",null,[t("mi",null,"u"),t("mi",null,"β")])]),t("mrow",null,[t("mi",null,"∂"),t("msup",null,[t("mi",null,"x"),t("mi",null,"α")])])]),t("mo",null,".")])],-1))]),s[43]||(s[43]=t("p",null,"Differentiable version.",-1)),s[44]||(s[44]=t("p",null,[t("a",{href:"https://github.com/agdestein/IncompressibleNavierStokes.jl/blob/a63d20e3a410f14ba6cc612a27e89f8974019d9f/src/operators.jl#L1425",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",E,[t("summary",null,[s[45]||(s[45]=t("a",{id:"IncompressibleNavierStokes.applybodyforce!-NTuple{4, Any}",href:"#IncompressibleNavierStokes.applybodyforce!-NTuple{4, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.applybodyforce!")],-1)),s[46]||(s[46]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[47]||(s[47]=a('
julia
applybodyforce!(F, u, t, setup) -> Any
Compute body force (in-place version). Add the result to F.
source
',3))]),t("details",M,[t("summary",null,[s[48]||(s[48]=t("a",{id:"IncompressibleNavierStokes.applybodyforce-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.applybodyforce-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.applybodyforce")],-1)),s[49]||(s[49]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[50]||(s[50]=a('
julia
applybodyforce(u, t, setup) -> Any
Compute body force (differentiable version).
source
',3))]),t("details",A,[t("summary",null,[s[51]||(s[51]=t("a",{id:"IncompressibleNavierStokes.applypressure!-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.applypressure!-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.applypressure!")],-1)),s[52]||(s[52]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[53]||(s[53]=a('
julia
applypressure!(u, p, setup) -> Any
Subtract pressure gradient (in-place version).
source
',3))]),t("details",C,[t("summary",null,[s[54]||(s[54]=t("a",{id:"IncompressibleNavierStokes.applypressure-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.applypressure-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.applypressure")],-1)),s[55]||(s[55]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[56]||(s[56]=a('
julia
applypressure(u, p, setup) -> Any
Subtract pressure gradient (differentiable version).
source
',3))]),t("details",V,[t("summary",null,[s[57]||(s[57]=t("a",{id:"IncompressibleNavierStokes.avg-NTuple{4, Any}",href:"#IncompressibleNavierStokes.avg-NTuple{4, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.avg")],-1)),s[58]||(s[58]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[59]||(s[59]=a('
julia
avg(ϕ, Δ, I, α) -> Any
Average scalar field ϕ in the α-direction.
source
',3))]),t("details",Z,[t("summary",null,[s[60]||(s[60]=t("a",{id:"IncompressibleNavierStokes.convection!-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.convection!-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.convection!")],-1)),s[61]||(s[61]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[62]||(s[62]=a('
julia
convection!(F, u, setup) -> Any
Compute convective term (in-place version). Add the result to F.
source
',3))]),t("details",S,[t("summary",null,[s[63]||(s[63]=t("a",{id:"IncompressibleNavierStokes.convection-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.convection-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.convection")],-1)),s[64]||(s[64]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[65]||(s[65]=a('
julia
convection(u, setup) -> Any
Compute convective term (differentiable version).
source
',3))]),t("details",N,[t("summary",null,[s[66]||(s[66]=t("a",{id:"IncompressibleNavierStokes.convection_diffusion_temp!-NTuple{4, Any}",href:"#IncompressibleNavierStokes.convection_diffusion_temp!-NTuple{4, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.convection_diffusion_temp!")],-1)),s[67]||(s[67]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[68]||(s[68]=a('
julia
convection_diffusion_temp!(c, u, temp, setup) -> Any
Compute convection-diffusion term for the temperature equation. (in-place version). Add result to c.
source
',3))]),t("details",I,[t("summary",null,[s[69]||(s[69]=t("a",{id:"IncompressibleNavierStokes.convection_diffusion_temp-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.convection_diffusion_temp-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.convection_diffusion_temp")],-1)),s[70]||(s[70]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[71]||(s[71]=a('
julia
convection_diffusion_temp(u, temp, setup) -> Any
Compute convection-diffusion term for the temperature equation. (differentiable version).
source
',3))]),t("details",D,[t("summary",null,[s[72]||(s[72]=t("a",{id:"IncompressibleNavierStokes.convectiondiffusion!-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.convectiondiffusion!-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.convectiondiffusion!")],-1)),s[73]||(s[73]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[74]||(s[74]=a('
julia
convectiondiffusion!(F, u, setup) -> Any
Compute convective and diffusive terms (in-place version). Add the result to F.
source
',3))]),t("details",F,[t("summary",null,[s[75]||(s[75]=t("a",{id:"IncompressibleNavierStokes.diffusion!-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.diffusion!-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.diffusion!")],-1)),s[76]||(s[76]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[77]||(s[77]=a('
julia
diffusion!(F, u, setup; use_viscosity) -> Any
Compute diffusive term (in-place version). Add the result to F.
Keyword arguments:
  • with_viscosity = true: Include viscosity in the operator.
source
',5))]),t("details",O,[t("summary",null,[s[78]||(s[78]=t("a",{id:"IncompressibleNavierStokes.diffusion-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.diffusion-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.diffusion")],-1)),s[79]||(s[79]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[80]||(s[80]=a('
julia
diffusion(u, setup; kwargs...) -> Any
Compute diffusive term (differentiable version).
source
',3))]),t("details",B,[t("summary",null,[s[81]||(s[81]=t("a",{id:"IncompressibleNavierStokes.dissipation!-NTuple{4, Any}",href:"#IncompressibleNavierStokes.dissipation!-NTuple{4, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.dissipation!")],-1)),s[82]||(s[82]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[83]||(s[83]=a('
julia
dissipation!(diss, diff, u, setup) -> Any
Compute dissipation term for the temperature equation (in-place version). Add result to diss.
source
',3))]),t("details",R,[t("summary",null,[s[84]||(s[84]=t("a",{id:"IncompressibleNavierStokes.dissipation-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.dissipation-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.dissipation")],-1)),s[85]||(s[85]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[86]||(s[86]=a('
julia
dissipation(u, setup) -> Any
Compute dissipation term for the temperature equation (differentiable version).
source
',3))]),t("details",G,[t("summary",null,[s[87]||(s[87]=t("a",{id:"IncompressibleNavierStokes.dissipation_from_strain!-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.dissipation_from_strain!-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.dissipation_from_strain!")],-1)),s[88]||(s[88]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[89]||(s[89]=a('
julia
dissipation_from_strain!(ϵ, u, setup) -> Any
Compute dissipation term from strain-rate tensor (in-place version).
source
',3))]),t("details",J,[t("summary",null,[s[90]||(s[90]=t("a",{id:"IncompressibleNavierStokes.dissipation_from_strain-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.dissipation_from_strain-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.dissipation_from_strain")],-1)),s[91]||(s[91]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[96]||(s[96]=a('
julia
dissipation_from_strain(u, setup) -> Any
',1)),t("p",null,[s[94]||(s[94]=e("Compute dissipation term ")),t("mjx-container",z,[(o(),n("svg",X,s[92]||(s[92]=[a('',1)]))),s[93]||(s[93]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mn",null,"2"),t("mi",null,"ν"),t("mo",{fence:"false",stretchy:"false"},"⟨"),t("msub",null,[t("mi",null,"S"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",null,"i"),t("mi",null,"j")])]),t("msub",null,[t("mi",null,"S"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",null,"i"),t("mi",null,"j")])]),t("mo",{fence:"false",stretchy:"false"},"⟩")])],-1))]),s[95]||(s[95]=e(" from strain-rate tensor (differentiable version)."))]),s[97]||(s[97]=t("p",null,[t("a",{href:"https://github.com/agdestein/IncompressibleNavierStokes.jl/blob/a63d20e3a410f14ba6cc612a27e89f8974019d9f/src/operators.jl#L810",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",q,[t("summary",null,[s[98]||(s[98]=t("a",{id:"IncompressibleNavierStokes.divergence!-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.divergence!-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.divergence!")],-1)),s[99]||(s[99]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[100]||(s[100]=a('
julia
divergence!(div, u, setup) -> Any
Compute divergence of velocity field (in-place version).
source
',3))]),t("details",P,[t("summary",null,[s[101]||(s[101]=t("a",{id:"IncompressibleNavierStokes.divergence-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.divergence-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.divergence")],-1)),s[102]||(s[102]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[103]||(s[103]=a('
julia
divergence(u, setup) -> Any
Compute divergence of velocity field (differentiable version).
source
',3))]),t("details",K,[t("summary",null,[s[104]||(s[104]=t("a",{id:"IncompressibleNavierStokes.divoftensor!-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.divoftensor!-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.divoftensor!")],-1)),s[105]||(s[105]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[106]||(s[106]=a('
julia
divoftensor!(s, σ, setup) -> Any
Compute divergence of a tensor with all components in the pressure points (in-place version). The stress tensors should be precomputed and stored in σ.
source
',3))]),t("details",U,[t("summary",null,[s[107]||(s[107]=t("a",{id:"IncompressibleNavierStokes.eig2field!-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.eig2field!-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.eig2field!")],-1)),s[108]||(s[108]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[113]||(s[113]=a('
julia
eig2field!(λ, u, setup) -> Any
',1)),t("p",null,[s[111]||(s[111]=e("Compute the second eigenvalue of ")),t("mjx-container",$,[(o(),n("svg",W,s[109]||(s[109]=[a('',1)]))),s[110]||(s[110]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msup",null,[t("mi",null,"S"),t("mn",null,"2")]),t("mo",null,"+"),t("msup",null,[t("mi",null,"R"),t("mn",null,"2")])])],-1))]),s[112]||(s[112]=e(" (in-place version)."))]),s[114]||(s[114]=t("p",null,[t("a",{href:"https://github.com/agdestein/IncompressibleNavierStokes.jl/blob/a63d20e3a410f14ba6cc612a27e89f8974019d9f/src/operators.jl#L1470",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",Y,[t("summary",null,[s[115]||(s[115]=t("a",{id:"IncompressibleNavierStokes.eig2field-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.eig2field-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.eig2field")],-1)),s[116]||(s[116]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[123]||(s[123]=a('
julia
eig2field(u, setup) -> Any
',1)),t("p",null,[s[119]||(s[119]=e("Compute the second eigenvalue of ")),t("mjx-container",_,[(o(),n("svg",t1,s[117]||(s[117]=[a('',1)]))),s[118]||(s[118]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msup",null,[t("mi",null,"S"),t("mn",null,"2")]),t("mo",null,"+"),t("msup",null,[t("mi",null,"R"),t("mn",null,"2")])])],-1))]),s[120]||(s[120]=e(", as proposed by Jeong and Hussain [")),s[121]||(s[121]=t("a",{href:"/IncompressibleNavierStokes.jl/dev/references#Jeong1995"},"6",-1)),s[122]||(s[122]=e("]."))]),s[124]||(s[124]=t("p",null,"Differentiable version.",-1)),s[125]||(s[125]=t("p",null,[t("a",{href:"https://github.com/agdestein/IncompressibleNavierStokes.jl/blob/a63d20e3a410f14ba6cc612a27e89f8974019d9f/src/operators.jl#L1462",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",s1,[t("summary",null,[s[126]||(s[126]=t("a",{id:"IncompressibleNavierStokes.get_scale_numbers-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.get_scale_numbers-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.get_scale_numbers")],-1)),s[127]||(s[127]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[149]||(s[149]=a(`
julia
get_scale_numbers(
     u,
     setup
-) -> NamedTuple{(:uavg, :ϵ, :η, :λ, :Reλ, :L, :τ), <:Tuple{Any, Any, Any, Any, Any, Nothing, Nothing}}
Get the following dimensional scale numbers [7]:
`,2)),t("ul",null,[t("li",null,[t("p",null,[s[130]||(s[130]=e("Velocity ")),t("mjx-container",e1,[(o(),n("svg",a1,s[128]||(s[128]=[a('',1)]))),s[129]||(s[129]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 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julia
gravity!(F, temp, setup) -> Any
Compute gravity term (in-place version). add the result to F.
source
',3))]),t("details",k1,[t("summary",null,[s[154]||(s[154]=t("a",{id:"IncompressibleNavierStokes.gravity-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.gravity-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.gravity")],-1)),s[155]||(s[155]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[156]||(s[156]=a('
julia
gravity(temp, setup) -> Any
Compute gravity term (differentiable version).
source
',3))]),t("details",c1,[t("summary",null,[s[157]||(s[157]=t("a",{id:"IncompressibleNavierStokes.interpolate_u_p!-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.interpolate_u_p!-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.interpolate_u_p!")],-1)),s[158]||(s[158]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[159]||(s[159]=a('
julia
interpolate_u_p!(up, u, setup) -> Any
Interpolate velocity to pressure points (in-place version).
source
',3))]),t("details",y1,[t("summary",null,[s[160]||(s[160]=t("a",{id:"IncompressibleNavierStokes.interpolate_u_p-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.interpolate_u_p-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.interpolate_u_p")],-1)),s[161]||(s[161]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[162]||(s[162]=a('
julia
interpolate_u_p(u, setup) -> Any
Interpolate velocity to pressure points (differentiable version).
source
',3))]),t("details",b1,[t("summary",null,[s[163]||(s[163]=t("a",{id:"IncompressibleNavierStokes.interpolate_ω_p!-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.interpolate_ω_p!-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.interpolate_ω_p!")],-1)),s[164]||(s[164]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[165]||(s[165]=a('
julia
interpolate_ω_p!(ωp, ω, setup) -> Any
Interpolate vorticity to pressure points (in-place version).
source
',3))]),t("details",f1,[t("summary",null,[s[166]||(s[166]=t("a",{id:"IncompressibleNavierStokes.interpolate_ω_p-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.interpolate_ω_p-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.interpolate_ω_p")],-1)),s[167]||(s[167]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[168]||(s[168]=a('
julia
interpolate_ω_p(ω, setup) -> Any
Interpolate vorticity to pressure points (differentiable version).
source
',3))]),t("details",v1,[t("summary",null,[s[169]||(s[169]=t("a",{id:"IncompressibleNavierStokes.kinetic_energy!-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.kinetic_energy!-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.kinetic_energy!")],-1)),s[170]||(s[170]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[171]||(s[171]=a('
julia
kinetic_energy!(ke, u, setup; interpolate_first) -> Any
Compute kinetic energy field (in-place version).
source
',3))]),t("details",x1,[t("summary",null,[s[172]||(s[172]=t("a",{id:"IncompressibleNavierStokes.kinetic_energy-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.kinetic_energy-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.kinetic_energy")],-1)),s[173]||(s[173]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[184]||(s[184]=a('
julia
kinetic_energy(u, setup; kwargs...) -> Any
',1)),t("p",null,[s[176]||(s[176]=e("Compute kinetic energy field ")),t("mjx-container",w1,[(o(),n("svg",j1,s[174]||(s[174]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D458",d:"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z",style:{"stroke-width":"3"}})])])],-1)]))),s[175]||(s[175]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"k")])],-1))]),s[177]||(s[177]=e(" (in-place version). 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julia
laplacian!(L, p, setup) -> Any
Compute Laplacian of pressure field (in-place version).
source
',3))]),t("details",C1,[t("summary",null,[s[192]||(s[192]=t("a",{id:"IncompressibleNavierStokes.laplacian-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.laplacian-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.laplacian")],-1)),s[193]||(s[193]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[194]||(s[194]=a('
julia
laplacian(p, setup) -> Any
Compute Laplacian of pressure field (differentiable version).
source
',3))]),t("details",V1,[t("summary",null,[s[195]||(s[195]=t("a",{id:"IncompressibleNavierStokes.momentum!-NTuple{5, Any}",href:"#IncompressibleNavierStokes.momentum!-NTuple{5, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.momentum!")],-1)),s[196]||(s[196]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[197]||(s[197]=a('
julia
momentum!(F, u, temp, t, setup) -> Any
Right hand side of momentum equations, excluding pressure gradient (in-place version).
source
',3))]),t("details",Z1,[t("summary",null,[s[198]||(s[198]=t("a",{id:"IncompressibleNavierStokes.momentum-NTuple{4, Any}",href:"#IncompressibleNavierStokes.momentum-NTuple{4, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.momentum")],-1)),s[199]||(s[199]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[200]||(s[200]=a('
julia
momentum(u, temp, t, setup) -> Any
Right hand side of momentum equations, excluding pressure gradient (differentiable version).
source
',3))]),t("details",S1,[t("summary",null,[s[201]||(s[201]=t("a",{id:"IncompressibleNavierStokes.pressuregradient!-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.pressuregradient!-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.pressuregradient!")],-1)),s[202]||(s[202]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[203]||(s[203]=a('
julia
pressuregradient!(G, p, setup) -> Any
Compute pressure gradient (in-place version).
source
',3))]),t("details",N1,[t("summary",null,[s[204]||(s[204]=t("a",{id:"IncompressibleNavierStokes.pressuregradient-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.pressuregradient-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.pressuregradient")],-1)),s[205]||(s[205]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[206]||(s[206]=a('
julia
pressuregradient(p, setup) -> Any
Compute pressure gradient (differentiable version).
source
',3))]),t("details",I1,[t("summary",null,[s[207]||(s[207]=t("a",{id:"IncompressibleNavierStokes.scalewithvolume!-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.scalewithvolume!-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.scalewithvolume!")],-1)),s[208]||(s[208]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[209]||(s[209]=a('
julia
scalewithvolume!(p, setup) -> Any
Scale scalar field with volume sizes (in-place version).
source
',3))]),t("details",D1,[t("summary",null,[s[210]||(s[210]=t("a",{id:"IncompressibleNavierStokes.scalewithvolume-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.scalewithvolume-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.scalewithvolume")],-1)),s[211]||(s[211]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[212]||(s[212]=a('
julia
scalewithvolume(p, setup) -> Any
Scale scalar field p with volume sizes (differentiable version).
source
',3))]),t("details",F1,[t("summary",null,[s[213]||(s[213]=t("a",{id:"IncompressibleNavierStokes.smagorinsky_closure-Tuple{Any}",href:"#IncompressibleNavierStokes.smagorinsky_closure-Tuple{Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.smagorinsky_closure")],-1)),s[214]||(s[214]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[215]||(s[215]=a(`
julia
smagorinsky_closure(
+) -> NamedTuple{(:uavg, :ϵ, :η, :λ, :Reλ, :L, :τ), <:Tuple{Any, Any, Any, Any, Any, Nothing, Nothing}}
Get the following dimensional scale numbers [7]:
`,2)),t("ul",null,[t("li",null,[t("p",null,[s[130]||(s[130]=e("Velocity ")),t("mjx-container",e1,[(o(),n("svg",a1,s[128]||(s[128]=[a('',1)]))),s[129]||(s[129]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 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")),t("mjx-container",h1,[(o(),n("svg",g1,s[146]||(s[146]=[a('',1)]))),s[147]||(s[147]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"τ"),t("mo",null,"="),t("mfrac",null,[t("mi",null,"L"),t("msub",null,[t("mi",null,"u"),t("mtext",null,"avg")])])])],-1))])])])]),s[150]||(s[150]=t("p",null,[t("a",{href:"https://github.com/agdestein/IncompressibleNavierStokes.jl/blob/a63d20e3a410f14ba6cc612a27e89f8974019d9f/src/operators.jl#L1558",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",u1,[t("summary",null,[s[151]||(s[151]=t("a",{id:"IncompressibleNavierStokes.gravity!-Tuple{Any, 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julia
gravity!(F, temp, setup) -> Any
Compute gravity term (in-place version). add the result to F.
source
',3))]),t("details",k1,[t("summary",null,[s[154]||(s[154]=t("a",{id:"IncompressibleNavierStokes.gravity-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.gravity-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.gravity")],-1)),s[155]||(s[155]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[156]||(s[156]=a('
julia
gravity(temp, setup) -> Any
Compute gravity term (differentiable version).
source
',3))]),t("details",c1,[t("summary",null,[s[157]||(s[157]=t("a",{id:"IncompressibleNavierStokes.interpolate_u_p!-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.interpolate_u_p!-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.interpolate_u_p!")],-1)),s[158]||(s[158]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[159]||(s[159]=a('
julia
interpolate_u_p!(up, u, setup) -> Any
Interpolate velocity to pressure points (in-place version).
source
',3))]),t("details",y1,[t("summary",null,[s[160]||(s[160]=t("a",{id:"IncompressibleNavierStokes.interpolate_u_p-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.interpolate_u_p-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.interpolate_u_p")],-1)),s[161]||(s[161]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[162]||(s[162]=a('
julia
interpolate_u_p(u, setup) -> Any
Interpolate velocity to pressure points (differentiable version).
source
',3))]),t("details",b1,[t("summary",null,[s[163]||(s[163]=t("a",{id:"IncompressibleNavierStokes.interpolate_ω_p!-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.interpolate_ω_p!-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.interpolate_ω_p!")],-1)),s[164]||(s[164]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[165]||(s[165]=a('
julia
interpolate_ω_p!(ωp, ω, setup) -> Any
Interpolate vorticity to pressure points (in-place version).
source
',3))]),t("details",f1,[t("summary",null,[s[166]||(s[166]=t("a",{id:"IncompressibleNavierStokes.interpolate_ω_p-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.interpolate_ω_p-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.interpolate_ω_p")],-1)),s[167]||(s[167]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[168]||(s[168]=a('
julia
interpolate_ω_p(ω, setup) -> Any
Interpolate vorticity to pressure points (differentiable version).
source
',3))]),t("details",v1,[t("summary",null,[s[169]||(s[169]=t("a",{id:"IncompressibleNavierStokes.kinetic_energy!-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.kinetic_energy!-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.kinetic_energy!")],-1)),s[170]||(s[170]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[171]||(s[171]=a('
julia
kinetic_energy!(ke, u, setup; interpolate_first) -> Any
Compute kinetic energy field (in-place version).
source
',3))]),t("details",x1,[t("summary",null,[s[172]||(s[172]=t("a",{id:"IncompressibleNavierStokes.kinetic_energy-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.kinetic_energy-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.kinetic_energy")],-1)),s[173]||(s[173]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[184]||(s[184]=a('
julia
kinetic_energy(u, setup; kwargs...) -> Any
',1)),t("p",null,[s[176]||(s[176]=e("Compute kinetic energy field ")),t("mjx-container",w1,[(o(),n("svg",j1,s[174]||(s[174]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D458",d:"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z",style:{"stroke-width":"3"}})])])],-1)]))),s[175]||(s[175]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"k")])],-1))]),s[177]||(s[177]=e(" (in-place version). If ")),s[178]||(s[178]=t("code",null,"interpolate_first",-1)),s[179]||(s[179]=e(" is true, it is given by"))]),t("mjx-container",H1,[(o(),n("svg",L1,s[180]||(s[180]=[a('',1)]))),s[181]||(s[181]=t("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[t("msub",null,[t("mi",null,"k"),t("mi",null,"I")]),t("mo",null,"="),t("mfrac",null,[t("mn",null,"1"),t("mn",null,"8")]),t("munder",null,[t("mo",{"data-mjx-texclass":"OP"},"∑"),t("mi",null,"α")]),t("mo",{stretchy:"false"},"("),t("msubsup",null,[t("mi",null,"u"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",null,"I"),t("mo",null,"+"),t("msub",null,[t("mi",null,"h"),t("mi",null,"α")])]),t("mi",null,"α")]),t("mo",null,"+"),t("msubsup",null,[t("mi",null,"u"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",null,"I"),t("mo",null,"−"),t("msub",null,[t("mi",null,"h"),t("mi",null,"α")])]),t("mi",null,"α")]),t("msup",null,[t("mo",{stretchy:"false"},")"),t("mn",null,"2")]),t("mo",null,".")])],-1))]),s[185]||(s[185]=t("p",null,"Otherwise, it is given by",-1)),t("mjx-container",E1,[(o(),n("svg",M1,s[182]||(s[182]=[a('',1)]))),s[183]||(s[183]=t("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[t("msub",null,[t("mi",null,"k"),t("mi",null,"I")]),t("mo",null,"="),t("mfrac",null,[t("mn",null,"1"),t("mn",null,"4")]),t("munder",null,[t("mo",{"data-mjx-texclass":"OP"},"∑"),t("mi",null,"α")]),t("mo",{stretchy:"false"},"("),t("mo",{stretchy:"false"},"("),t("msubsup",null,[t("mi",null,"u"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",null,"I"),t("mo",null,"+"),t("msub",null,[t("mi",null,"h"),t("mi",null,"α")])]),t("mi",null,"α")]),t("msup",null,[t("mo",{stretchy:"false"},")"),t("mn",null,"2")]),t("mo",null,"+"),t("mo",{stretchy:"false"},"("),t("msubsup",null,[t("mi",null,"u"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",null,"I"),t("mo",null,"−"),t("msub",null,[t("mi",null,"h"),t("mi",null,"α")])]),t("mi",null,"α")]),t("msup",null,[t("mo",{stretchy:"false"},")"),t("mn",null,"2")]),t("mo",{stretchy:"false"},")"),t("mo",null,",")])],-1))]),s[186]||(s[186]=t("p",null,[e("as in ["),t("a",{href:"/IncompressibleNavierStokes.jl/dev/references#Sanderse2023"},"8"),e("].")],-1)),s[187]||(s[187]=t("p",null,"Differentiable version.",-1)),s[188]||(s[188]=t("p",null,[t("a",{href:"https://github.com/agdestein/IncompressibleNavierStokes.jl/blob/a63d20e3a410f14ba6cc612a27e89f8974019d9f/src/operators.jl#L1491",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",A1,[t("summary",null,[s[189]||(s[189]=t("a",{id:"IncompressibleNavierStokes.laplacian!-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.laplacian!-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.laplacian!")],-1)),s[190]||(s[190]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[191]||(s[191]=a('
julia
laplacian!(L, p, setup) -> Any
Compute Laplacian of pressure field (in-place version).
source
',3))]),t("details",C1,[t("summary",null,[s[192]||(s[192]=t("a",{id:"IncompressibleNavierStokes.laplacian-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.laplacian-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.laplacian")],-1)),s[193]||(s[193]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[194]||(s[194]=a('
julia
laplacian(p, setup) -> Any
Compute Laplacian of pressure field (differentiable version).
source
',3))]),t("details",V1,[t("summary",null,[s[195]||(s[195]=t("a",{id:"IncompressibleNavierStokes.momentum!-NTuple{5, Any}",href:"#IncompressibleNavierStokes.momentum!-NTuple{5, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.momentum!")],-1)),s[196]||(s[196]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[197]||(s[197]=a('
julia
momentum!(F, u, temp, t, setup) -> Any
Right hand side of momentum equations, excluding pressure gradient (in-place version).
source
',3))]),t("details",Z1,[t("summary",null,[s[198]||(s[198]=t("a",{id:"IncompressibleNavierStokes.momentum-NTuple{4, Any}",href:"#IncompressibleNavierStokes.momentum-NTuple{4, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.momentum")],-1)),s[199]||(s[199]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[200]||(s[200]=a('
julia
momentum(u, temp, t, setup) -> Any
Right hand side of momentum equations, excluding pressure gradient (differentiable version).
source
',3))]),t("details",S1,[t("summary",null,[s[201]||(s[201]=t("a",{id:"IncompressibleNavierStokes.pressuregradient!-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.pressuregradient!-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.pressuregradient!")],-1)),s[202]||(s[202]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[203]||(s[203]=a('
julia
pressuregradient!(G, p, setup) -> Any
Compute pressure gradient (in-place version).
source
',3))]),t("details",N1,[t("summary",null,[s[204]||(s[204]=t("a",{id:"IncompressibleNavierStokes.pressuregradient-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.pressuregradient-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.pressuregradient")],-1)),s[205]||(s[205]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[206]||(s[206]=a('
julia
pressuregradient(p, setup) -> Any
Compute pressure gradient (differentiable version).
source
',3))]),t("details",I1,[t("summary",null,[s[207]||(s[207]=t("a",{id:"IncompressibleNavierStokes.scalewithvolume!-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.scalewithvolume!-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.scalewithvolume!")],-1)),s[208]||(s[208]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[209]||(s[209]=a('
julia
scalewithvolume!(p, setup) -> Any
Scale scalar field with volume sizes (in-place version).
source
',3))]),t("details",D1,[t("summary",null,[s[210]||(s[210]=t("a",{id:"IncompressibleNavierStokes.scalewithvolume-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.scalewithvolume-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.scalewithvolume")],-1)),s[211]||(s[211]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[212]||(s[212]=a('
julia
scalewithvolume(p, setup) -> Any
Scale scalar field p with volume sizes (differentiable version).
source
',3))]),t("details",F1,[t("summary",null,[s[213]||(s[213]=t("a",{id:"IncompressibleNavierStokes.smagorinsky_closure-Tuple{Any}",href:"#IncompressibleNavierStokes.smagorinsky_closure-Tuple{Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.smagorinsky_closure")],-1)),s[214]||(s[214]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[215]||(s[215]=a(`
julia
smagorinsky_closure(
     setup
-) -> IncompressibleNavierStokes.var"#closure#188"
Create Smagorinsky closure model m. The model is called as m(u, θ), where the Smagorinsky constant θ should be a scalar between 0 and 1 (for example θ = 0.1).
source
`,3))]),t("details",O1,[t("summary",null,[s[216]||(s[216]=t("a",{id:"IncompressibleNavierStokes.smagtensor!-NTuple{4, Any}",href:"#IncompressibleNavierStokes.smagtensor!-NTuple{4, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.smagtensor!")],-1)),s[217]||(s[217]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[218]||(s[218]=a('
julia
smagtensor!(σ, u, θ, setup) -> Any
Compute Smagorinsky stress tensors σ[I] (in-place version). The Smagorinsky constant θ should be a scalar between 0 and 1.
source
',3))]),t("details",B1,[t("summary",null,[s[219]||(s[219]=t("a",{id:"IncompressibleNavierStokes.total_kinetic_energy-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.total_kinetic_energy-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.total_kinetic_energy")],-1)),s[220]||(s[220]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[221]||(s[221]=a('
julia
total_kinetic_energy(u, setup; kwargs...) -> Any
Compute total kinetic energy. The velocity components are interpolated to the volume centers and squared.
source
',3))]),t("details",R1,[t("summary",null,[s[222]||(s[222]=t("a",{id:"IncompressibleNavierStokes.unit_cartesian_indices-Tuple{Any}",href:"#IncompressibleNavierStokes.unit_cartesian_indices-Tuple{Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.unit_cartesian_indices")],-1)),s[223]||(s[223]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[224]||(s[224]=a('
julia
unit_cartesian_indices(D) -> Any
Get tuple of all unit vectors as Cartesian indices.
source
',3))]),t("details",G1,[t("summary",null,[s[225]||(s[225]=t("a",{id:"IncompressibleNavierStokes.vorticity!-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.vorticity!-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.vorticity!")],-1)),s[226]||(s[226]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[227]||(s[227]=a('
julia
vorticity!(ω, u, setup) -> Any
Compute vorticity field (in-place version).
source
',3))]),t("details",J1,[t("summary",null,[s[228]||(s[228]=t("a",{id:"IncompressibleNavierStokes.vorticity-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.vorticity-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.vorticity")],-1)),s[229]||(s[229]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[230]||(s[230]=a('
julia
vorticity(u, setup) -> Any
Compute vorticity field (differentiable version).
source
',3))]),t("details",z1,[t("summary",null,[s[231]||(s[231]=t("a",{id:"IncompressibleNavierStokes.lastdimcontract-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.lastdimcontract-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.lastdimcontract")],-1)),s[232]||(s[232]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[233]||(s[233]=a('
julia
lastdimcontract(a, b, setup) -> Any
Compute c[I] = sum_i a[I, i] * b[I, i], where c[:, i] and b[:, i] are tensor fields (elements are SMatrixes), a[:, i] is a scalar field, and the i dimension is contracted (multiple channels).
source
',3))]),t("details",X1,[t("summary",null,[s[234]||(s[234]=t("a",{id:"IncompressibleNavierStokes.tensorbasis!-NTuple{4, Any}",href:"#IncompressibleNavierStokes.tensorbasis!-NTuple{4, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.tensorbasis!")],-1)),s[235]||(s[235]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[257]||(s[257]=a('
julia
tensorbasis!(B, V, u, setup) -> Tuple{Any, Any}
',1)),t("p",null,[s[240]||(s[240]=e("Compute symmetry tensor basis ")),s[241]||(s[241]=t("code",null,"B[1]",-1)),s[242]||(s[242]=e("-")),s[243]||(s[243]=t("code",null,"B[11]",-1)),s[244]||(s[244]=e(" and invariants ")),s[245]||(s[245]=t("code",null,"V[1]",-1)),s[246]||(s[246]=e("-")),s[247]||(s[247]=t("code",null,"V[5]",-1)),s[248]||(s[248]=e(", as specified in [")),s[249]||(s[249]=t("a",{href:"/IncompressibleNavierStokes.jl/dev/references#Silvis2017"},"9",-1)),s[250]||(s[250]=e("] in equations (9) and (11). Note that ")),s[251]||(s[251]=t("code",null,"B[1]",-1)),s[252]||(s[252]=e(" corresponds to ")),t("mjx-container",q1,[(o(),n("svg",P1,s[236]||(s[236]=[a('',1)]))),s[237]||(s[237]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"T"),t("mn",null,"0")])])],-1))]),s[253]||(s[253]=e(" in the paper, and ")),s[254]||(s[254]=t("code",null,"V",-1)),s[255]||(s[255]=e(" to ")),t("mjx-container",K1,[(o(),n("svg",U1,s[238]||(s[238]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D43C",d:"M43 1Q26 1 26 10Q26 12 29 24Q34 43 39 45Q42 46 54 46H60Q120 46 136 53Q137 53 138 54Q143 56 149 77T198 273Q210 318 216 344Q286 624 286 626Q284 630 284 631Q274 637 213 637H193Q184 643 189 662Q193 677 195 680T209 683H213Q285 681 359 681Q481 681 487 683H497Q504 676 504 672T501 655T494 639Q491 637 471 637Q440 637 407 634Q393 631 388 623Q381 609 337 432Q326 385 315 341Q245 65 245 59Q245 52 255 50T307 46H339Q345 38 345 37T342 19Q338 6 332 0H316Q279 2 179 2Q143 2 113 2T65 2T43 1Z",style:{"stroke-width":"3"}})])])],-1)]))),s[239]||(s[239]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"I")])],-1))]),s[256]||(s[256]=e("."))]),s[258]||(s[258]=t("p",null,[t("a",{href:"https://github.com/agdestein/IncompressibleNavierStokes.jl/blob/03acafa3566742b1f37d722d2207f5dc21d031c0/src/tensorbasis.jl#L16",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",$1,[t("summary",null,[s[259]||(s[259]=t("a",{id:"IncompressibleNavierStokes.tensorbasis-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.tensorbasis-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.tensorbasis")],-1)),s[260]||(s[260]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[261]||(s[261]=a('
julia
tensorbasis(u, setup) -> Tuple{Any, Any}
Compute symmetry tensor basis (differentiable version).
source
',3))])])}const i2=Q(T,[["render",W1]]);export{l2 as __pageData,i2 as default};
+) -> IncompressibleNavierStokes.var"#closure#188"
Create Smagorinsky closure model m. The model is called as m(u, θ), where the Smagorinsky constant θ should be a scalar between 0 and 1 (for example θ = 0.1).
source
`,3))]),t("details",O1,[t("summary",null,[s[216]||(s[216]=t("a",{id:"IncompressibleNavierStokes.smagtensor!-NTuple{4, Any}",href:"#IncompressibleNavierStokes.smagtensor!-NTuple{4, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.smagtensor!")],-1)),s[217]||(s[217]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[218]||(s[218]=a('
julia
smagtensor!(σ, u, θ, setup) -> Any
Compute Smagorinsky stress tensors σ[I] (in-place version). The Smagorinsky constant θ should be a scalar between 0 and 1.
source
',3))]),t("details",B1,[t("summary",null,[s[219]||(s[219]=t("a",{id:"IncompressibleNavierStokes.total_kinetic_energy-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.total_kinetic_energy-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.total_kinetic_energy")],-1)),s[220]||(s[220]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[221]||(s[221]=a('
julia
total_kinetic_energy(u, setup; kwargs...) -> Any
Compute total kinetic energy. The velocity components are interpolated to the volume centers and squared.
source
',3))]),t("details",R1,[t("summary",null,[s[222]||(s[222]=t("a",{id:"IncompressibleNavierStokes.unit_cartesian_indices-Tuple{Any}",href:"#IncompressibleNavierStokes.unit_cartesian_indices-Tuple{Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.unit_cartesian_indices")],-1)),s[223]||(s[223]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[224]||(s[224]=a('
julia
unit_cartesian_indices(D) -> Any
Get tuple of all unit vectors as Cartesian indices.
source
',3))]),t("details",G1,[t("summary",null,[s[225]||(s[225]=t("a",{id:"IncompressibleNavierStokes.vorticity!-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.vorticity!-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.vorticity!")],-1)),s[226]||(s[226]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[227]||(s[227]=a('
julia
vorticity!(ω, u, setup) -> Any
Compute vorticity field (in-place version).
source
',3))]),t("details",J1,[t("summary",null,[s[228]||(s[228]=t("a",{id:"IncompressibleNavierStokes.vorticity-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.vorticity-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.vorticity")],-1)),s[229]||(s[229]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[230]||(s[230]=a('
julia
vorticity(u, setup) -> Any
Compute vorticity field (differentiable version).
source
',3))]),t("details",z1,[t("summary",null,[s[231]||(s[231]=t("a",{id:"IncompressibleNavierStokes.lastdimcontract-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.lastdimcontract-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.lastdimcontract")],-1)),s[232]||(s[232]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[233]||(s[233]=a('
julia
lastdimcontract(a, b, setup) -> Any
Compute c[I] = sum_i a[I, i] * b[I, i], where c[:, i] and b[:, i] are tensor fields (elements are SMatrixes), a[:, i] is a scalar field, and the i dimension is contracted (multiple channels).
source
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julia
tensorbasis!(B, V, u, setup) -> Tuple{Any, Any}
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julia
tensorbasis(u, setup) -> Tuple{Any, Any}
Compute symmetry tensor basis (differentiable version).
source
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similarity index 97%
rename from dev/assets/manual_operators.md.d5WWXEo-.js
rename to dev/assets/manual_operators.md.DDNX2oaK.lean.js
index eee6825e6..353a61433 100644
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Operators ​
All discrete operators are built using KernelAbstractions.jl and Cartesian indices, similar to WaterLily.jl. This allows for dimension- and backend-agnostic code. See this blog post for how to write kernels. IncompressibleNavierStokes previously relied on assembling sparse operators to perform the same operations. While being very efficient and also compatible with CUDA (CUSPARSE), storing these matrices in memory is expensive for large 3D problems.
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julia
struct Offset{D}
Cartesian index unit vector in D = 2 or D = 3 dimensions. Calling Offset(D)(α) returns a Cartesian index with 1 in the dimension α and zeros elsewhere.
See https://b-fg.github.io/2023/05/07/waterlily-on-gpu.html for writing kernel loops using Cartesian indices.
Fields
source
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julia
Dfield!(d, G, p, setup; ϵ) -> Any
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julia
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julia
Qfield!(Q, u, setup) -> Any
',1)),t("p",null,[s[28]||(s[28]=e("Compute the ")),t("mjx-container",f,[(o(),n("svg",v,s[26]||(s[26]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D444",d:"M399 -80Q399 -47 400 -30T402 -11V-7L387 -11Q341 -22 303 -22Q208 -22 138 35T51 201Q50 209 50 244Q50 346 98 438T227 601Q351 704 476 704Q514 704 524 703Q621 689 680 617T740 435Q740 255 592 107Q529 47 461 16L444 8V3Q444 2 449 -24T470 -66T516 -82Q551 -82 583 -60T625 -3Q631 11 638 11Q647 11 649 2Q649 -6 639 -34T611 -100T557 -165T481 -194Q399 -194 399 -87V-80ZM636 468Q636 523 621 564T580 625T530 655T477 665Q429 665 379 640Q277 591 215 464T153 216Q153 110 207 59Q231 38 236 38V46Q236 86 269 120T347 155Q372 155 390 144T417 114T429 82T435 55L448 64Q512 108 557 185T619 334T636 468ZM314 18Q362 18 404 39L403 49Q399 104 366 115Q354 117 347 117Q344 117 341 117T337 118Q317 118 296 98T274 52Q274 18 314 18Z",style:{"stroke-width":"3"}})])])],-1)]))),s[27]||(s[27]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"Q")])],-1))]),s[29]||(s[29]=e("-field (in-place version)."))]),s[31]||(s[31]=t("p",null,[t("a",{href:"https://github.com/agdestein/IncompressibleNavierStokes.jl/blob/03acafa3566742b1f37d722d2207f5dc21d031c0/src/operators.jl#L1440",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",x,[t("summary",null,[s[32]||(s[32]=t("a",{id:"IncompressibleNavierStokes.Qfield-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.Qfield-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.Qfield")],-1)),s[33]||(s[33]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[42]||(s[42]=a('
julia
Qfield(u, setup) -> Any
',1)),t("p",null,[s[36]||(s[36]=e("Compute ")),t("mjx-container",w,[(o(),n("svg",j,s[34]||(s[34]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D444",d:"M399 -80Q399 -47 400 -30T402 -11V-7L387 -11Q341 -22 303 -22Q208 -22 138 35T51 201Q50 209 50 244Q50 346 98 438T227 601Q351 704 476 704Q514 704 524 703Q621 689 680 617T740 435Q740 255 592 107Q529 47 461 16L444 8V3Q444 2 449 -24T470 -66T516 -82Q551 -82 583 -60T625 -3Q631 11 638 11Q647 11 649 2Q649 -6 639 -34T611 -100T557 -165T481 -194Q399 -194 399 -87V-80ZM636 468Q636 523 621 564T580 625T530 655T477 665Q429 665 379 640Q277 591 215 464T153 216Q153 110 207 59Q231 38 236 38V46Q236 86 269 120T347 155Q372 155 390 144T417 114T429 82T435 55L448 64Q512 108 557 185T619 334T636 468ZM314 18Q362 18 404 39L403 49Q399 104 366 115Q354 117 347 117Q344 117 341 117T337 118Q317 118 296 98T274 52Q274 18 314 18Z",style:{"stroke-width":"3"}})])])],-1)]))),s[35]||(s[35]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"Q")])],-1))]),s[37]||(s[37]=e("-field [")),s[38]||(s[38]=t("a",{href:"/IncompressibleNavierStokes.jl/dev/references#Jeong1995"},"6",-1)),s[39]||(s[39]=e("] given by"))]),t("mjx-container",H,[(o(),n("svg",L,s[40]||(s[40]=[a('',1)]))),s[41]||(s[41]=t("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[t("mi",null,"Q"),t("mo",null,"="),t("mo",null,"−"),t("mfrac",null,[t("mn",null,"1"),t("mn",null,"2")]),t("munder",null,[t("mo",{"data-mjx-texclass":"OP"},"∑"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",null,"α"),t("mo",null,","),t("mi",null,"β")])]),t("mfrac",null,[t("mrow",null,[t("mi",null,"∂"),t("msup",null,[t("mi",null,"u"),t("mi",null,"α")])]),t("mrow",null,[t("mi",null,"∂"),t("msup",null,[t("mi",null,"x"),t("mi",null,"β")])])]),t("mfrac",null,[t("mrow",null,[t("mi",null,"∂"),t("msup",null,[t("mi",null,"u"),t("mi",null,"β")])]),t("mrow",null,[t("mi",null,"∂"),t("msup",null,[t("mi",null,"x"),t("mi",null,"α")])])]),t("mo",null,".")])],-1))]),s[43]||(s[43]=t("p",null,"Differentiable version.",-1)),s[44]||(s[44]=t("p",null,[t("a",{href:"https://github.com/agdestein/IncompressibleNavierStokes.jl/blob/03acafa3566742b1f37d722d2207f5dc21d031c0/src/operators.jl#L1425",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",E,[t("summary",null,[s[45]||(s[45]=t("a",{id:"IncompressibleNavierStokes.applybodyforce!-NTuple{4, Any}",href:"#IncompressibleNavierStokes.applybodyforce!-NTuple{4, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.applybodyforce!")],-1)),s[46]||(s[46]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[47]||(s[47]=a('
julia
applybodyforce!(F, u, t, setup) -> Any
Compute body force (in-place version). Add the result to F.
source
',3))]),t("details",M,[t("summary",null,[s[48]||(s[48]=t("a",{id:"IncompressibleNavierStokes.applybodyforce-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.applybodyforce-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.applybodyforce")],-1)),s[49]||(s[49]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[50]||(s[50]=a('
julia
applybodyforce(u, t, setup) -> Any
Compute body force (differentiable version).
source
',3))]),t("details",A,[t("summary",null,[s[51]||(s[51]=t("a",{id:"IncompressibleNavierStokes.applypressure!-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.applypressure!-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.applypressure!")],-1)),s[52]||(s[52]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[53]||(s[53]=a('
julia
applypressure!(u, p, setup) -> Any
Subtract pressure gradient (in-place version).
source
',3))]),t("details",C,[t("summary",null,[s[54]||(s[54]=t("a",{id:"IncompressibleNavierStokes.applypressure-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.applypressure-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.applypressure")],-1)),s[55]||(s[55]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[56]||(s[56]=a('
julia
applypressure(u, p, setup) -> Any
Subtract pressure gradient (differentiable version).
source
',3))]),t("details",V,[t("summary",null,[s[57]||(s[57]=t("a",{id:"IncompressibleNavierStokes.avg-NTuple{4, Any}",href:"#IncompressibleNavierStokes.avg-NTuple{4, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.avg")],-1)),s[58]||(s[58]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[59]||(s[59]=a('
julia
avg(ϕ, Δ, I, α) -> Any
Average scalar field ϕ in the α-direction.
source
',3))]),t("details",Z,[t("summary",null,[s[60]||(s[60]=t("a",{id:"IncompressibleNavierStokes.convection!-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.convection!-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.convection!")],-1)),s[61]||(s[61]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[62]||(s[62]=a('
julia
convection!(F, u, setup) -> Any
Compute convective term (in-place version). Add the result to F.
source
',3))]),t("details",S,[t("summary",null,[s[63]||(s[63]=t("a",{id:"IncompressibleNavierStokes.convection-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.convection-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.convection")],-1)),s[64]||(s[64]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[65]||(s[65]=a('
julia
convection(u, setup) -> Any
Compute convective term (differentiable version).
source
',3))]),t("details",N,[t("summary",null,[s[66]||(s[66]=t("a",{id:"IncompressibleNavierStokes.convection_diffusion_temp!-NTuple{4, Any}",href:"#IncompressibleNavierStokes.convection_diffusion_temp!-NTuple{4, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.convection_diffusion_temp!")],-1)),s[67]||(s[67]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[68]||(s[68]=a('
julia
convection_diffusion_temp!(c, u, temp, setup) -> Any
Compute convection-diffusion term for the temperature equation. (in-place version). Add result to c.
source
',3))]),t("details",I,[t("summary",null,[s[69]||(s[69]=t("a",{id:"IncompressibleNavierStokes.convection_diffusion_temp-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.convection_diffusion_temp-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.convection_diffusion_temp")],-1)),s[70]||(s[70]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[71]||(s[71]=a('
julia
convection_diffusion_temp(u, temp, setup) -> Any
Compute convection-diffusion term for the temperature equation. (differentiable version).
source
',3))]),t("details",D,[t("summary",null,[s[72]||(s[72]=t("a",{id:"IncompressibleNavierStokes.convectiondiffusion!-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.convectiondiffusion!-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.convectiondiffusion!")],-1)),s[73]||(s[73]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[74]||(s[74]=a('
julia
convectiondiffusion!(F, u, setup) -> Any
Compute convective and diffusive terms (in-place version). Add the result to F.
source
',3))]),t("details",F,[t("summary",null,[s[75]||(s[75]=t("a",{id:"IncompressibleNavierStokes.diffusion!-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.diffusion!-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.diffusion!")],-1)),s[76]||(s[76]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[77]||(s[77]=a('
julia
diffusion!(F, u, setup; use_viscosity) -> Any
Compute diffusive term (in-place version). Add the result to F.
Keyword arguments:
  • with_viscosity = true: Include viscosity in the operator.
source
',5))]),t("details",O,[t("summary",null,[s[78]||(s[78]=t("a",{id:"IncompressibleNavierStokes.diffusion-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.diffusion-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.diffusion")],-1)),s[79]||(s[79]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[80]||(s[80]=a('
julia
diffusion(u, setup; kwargs...) -> Any
Compute diffusive term (differentiable version).
source
',3))]),t("details",B,[t("summary",null,[s[81]||(s[81]=t("a",{id:"IncompressibleNavierStokes.dissipation!-NTuple{4, Any}",href:"#IncompressibleNavierStokes.dissipation!-NTuple{4, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.dissipation!")],-1)),s[82]||(s[82]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[83]||(s[83]=a('
julia
dissipation!(diss, diff, u, setup) -> Any
Compute dissipation term for the temperature equation (in-place version). Add result to diss.
source
',3))]),t("details",R,[t("summary",null,[s[84]||(s[84]=t("a",{id:"IncompressibleNavierStokes.dissipation-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.dissipation-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.dissipation")],-1)),s[85]||(s[85]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[86]||(s[86]=a('
julia
dissipation(u, setup) -> Any
Compute dissipation term for the temperature equation (differentiable version).
source
',3))]),t("details",G,[t("summary",null,[s[87]||(s[87]=t("a",{id:"IncompressibleNavierStokes.dissipation_from_strain!-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.dissipation_from_strain!-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.dissipation_from_strain!")],-1)),s[88]||(s[88]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[89]||(s[89]=a('
julia
dissipation_from_strain!(ϵ, u, setup) -> Any
Compute dissipation term from strain-rate tensor (in-place version).
source
',3))]),t("details",J,[t("summary",null,[s[90]||(s[90]=t("a",{id:"IncompressibleNavierStokes.dissipation_from_strain-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.dissipation_from_strain-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.dissipation_from_strain")],-1)),s[91]||(s[91]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[96]||(s[96]=a('
julia
dissipation_from_strain(u, setup) -> Any
',1)),t("p",null,[s[94]||(s[94]=e("Compute dissipation term ")),t("mjx-container",z,[(o(),n("svg",X,s[92]||(s[92]=[a('',1)]))),s[93]||(s[93]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mn",null,"2"),t("mi",null,"ν"),t("mo",{fence:"false",stretchy:"false"},"⟨"),t("msub",null,[t("mi",null,"S"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",null,"i"),t("mi",null,"j")])]),t("msub",null,[t("mi",null,"S"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",null,"i"),t("mi",null,"j")])]),t("mo",{fence:"false",stretchy:"false"},"⟩")])],-1))]),s[95]||(s[95]=e(" from strain-rate tensor (differentiable version)."))]),s[97]||(s[97]=t("p",null,[t("a",{href:"https://github.com/agdestein/IncompressibleNavierStokes.jl/blob/03acafa3566742b1f37d722d2207f5dc21d031c0/src/operators.jl#L810",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",q,[t("summary",null,[s[98]||(s[98]=t("a",{id:"IncompressibleNavierStokes.divergence!-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.divergence!-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.divergence!")],-1)),s[99]||(s[99]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[100]||(s[100]=a('
julia
divergence!(div, u, setup) -> Any
Compute divergence of velocity field (in-place version).
source
',3))]),t("details",P,[t("summary",null,[s[101]||(s[101]=t("a",{id:"IncompressibleNavierStokes.divergence-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.divergence-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.divergence")],-1)),s[102]||(s[102]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[103]||(s[103]=a('
julia
divergence(u, setup) -> Any
Compute divergence of velocity field (differentiable version).
source
',3))]),t("details",K,[t("summary",null,[s[104]||(s[104]=t("a",{id:"IncompressibleNavierStokes.divoftensor!-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.divoftensor!-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.divoftensor!")],-1)),s[105]||(s[105]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[106]||(s[106]=a('
julia
divoftensor!(s, σ, setup) -> Any
Compute divergence of a tensor with all components in the pressure points (in-place version). The stress tensors should be precomputed and stored in σ.
source
',3))]),t("details",U,[t("summary",null,[s[107]||(s[107]=t("a",{id:"IncompressibleNavierStokes.eig2field!-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.eig2field!-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.eig2field!")],-1)),s[108]||(s[108]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[113]||(s[113]=a('
julia
eig2field!(λ, u, setup) -> Any
',1)),t("p",null,[s[111]||(s[111]=e("Compute the second eigenvalue of ")),t("mjx-container",$,[(o(),n("svg",W,s[109]||(s[109]=[a('',1)]))),s[110]||(s[110]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msup",null,[t("mi",null,"S"),t("mn",null,"2")]),t("mo",null,"+"),t("msup",null,[t("mi",null,"R"),t("mn",null,"2")])])],-1))]),s[112]||(s[112]=e(" (in-place version)."))]),s[114]||(s[114]=t("p",null,[t("a",{href:"https://github.com/agdestein/IncompressibleNavierStokes.jl/blob/03acafa3566742b1f37d722d2207f5dc21d031c0/src/operators.jl#L1470",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",Y,[t("summary",null,[s[115]||(s[115]=t("a",{id:"IncompressibleNavierStokes.eig2field-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.eig2field-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.eig2field")],-1)),s[116]||(s[116]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[123]||(s[123]=a('
julia
eig2field(u, setup) -> Any
',1)),t("p",null,[s[119]||(s[119]=e("Compute the second eigenvalue of ")),t("mjx-container",_,[(o(),n("svg",t1,s[117]||(s[117]=[a('',1)]))),s[118]||(s[118]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msup",null,[t("mi",null,"S"),t("mn",null,"2")]),t("mo",null,"+"),t("msup",null,[t("mi",null,"R"),t("mn",null,"2")])])],-1))]),s[120]||(s[120]=e(", as proposed by Jeong and Hussain [")),s[121]||(s[121]=t("a",{href:"/IncompressibleNavierStokes.jl/dev/references#Jeong1995"},"6",-1)),s[122]||(s[122]=e("]."))]),s[124]||(s[124]=t("p",null,"Differentiable version.",-1)),s[125]||(s[125]=t("p",null,[t("a",{href:"https://github.com/agdestein/IncompressibleNavierStokes.jl/blob/03acafa3566742b1f37d722d2207f5dc21d031c0/src/operators.jl#L1462",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",s1,[t("summary",null,[s[126]||(s[126]=t("a",{id:"IncompressibleNavierStokes.get_scale_numbers-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.get_scale_numbers-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.get_scale_numbers")],-1)),s[127]||(s[127]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[149]||(s[149]=a(`
julia
get_scale_numbers(
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Operators ​
All discrete operators are built using KernelAbstractions.jl and Cartesian indices, similar to WaterLily.jl. This allows for dimension- and backend-agnostic code. See this blog post for how to write kernels. IncompressibleNavierStokes previously relied on assembling sparse operators to perform the same operations. While being very efficient and also compatible with CUDA (CUSPARSE), storing these matrices in memory is expensive for large 3D problems.
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julia
struct Offset{D}
Cartesian index unit vector in D = 2 or D = 3 dimensions. Calling Offset(D)(α) returns a Cartesian index with 1 in the dimension α and zeros elsewhere.
See https://b-fg.github.io/2023/05/07/waterlily-on-gpu.html for writing kernel loops using Cartesian indices.
Fields
source
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julia
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julia
Qfield(u, setup) -> Any
',1)),t("p",null,[s[36]||(s[36]=e("Compute ")),t("mjx-container",w,[(o(),n("svg",j,s[34]||(s[34]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D444",d:"M399 -80Q399 -47 400 -30T402 -11V-7L387 -11Q341 -22 303 -22Q208 -22 138 35T51 201Q50 209 50 244Q50 346 98 438T227 601Q351 704 476 704Q514 704 524 703Q621 689 680 617T740 435Q740 255 592 107Q529 47 461 16L444 8V3Q444 2 449 -24T470 -66T516 -82Q551 -82 583 -60T625 -3Q631 11 638 11Q647 11 649 2Q649 -6 639 -34T611 -100T557 -165T481 -194Q399 -194 399 -87V-80ZM636 468Q636 523 621 564T580 625T530 655T477 665Q429 665 379 640Q277 591 215 464T153 216Q153 110 207 59Q231 38 236 38V46Q236 86 269 120T347 155Q372 155 390 144T417 114T429 82T435 55L448 64Q512 108 557 185T619 334T636 468ZM314 18Q362 18 404 39L403 49Q399 104 366 115Q354 117 347 117Q344 117 341 117T337 118Q317 118 296 98T274 52Q274 18 314 18Z",style:{"stroke-width":"3"}})])])],-1)]))),s[35]||(s[35]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"Q")])],-1))]),s[37]||(s[37]=e("-field [")),s[38]||(s[38]=t("a",{href:"/IncompressibleNavierStokes.jl/dev/references#Jeong1995"},"6",-1)),s[39]||(s[39]=e("] given by"))]),t("mjx-container",H,[(o(),n("svg",L,s[40]||(s[40]=[a('',1)]))),s[41]||(s[41]=t("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[t("mi",null,"Q"),t("mo",null,"="),t("mo",null,"−"),t("mfrac",null,[t("mn",null,"1"),t("mn",null,"2")]),t("munder",null,[t("mo",{"data-mjx-texclass":"OP"},"∑"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",null,"α"),t("mo",null,","),t("mi",null,"β")])]),t("mfrac",null,[t("mrow",null,[t("mi",null,"∂"),t("msup",null,[t("mi",null,"u"),t("mi",null,"α")])]),t("mrow",null,[t("mi",null,"∂"),t("msup",null,[t("mi",null,"x"),t("mi",null,"β")])])]),t("mfrac",null,[t("mrow",null,[t("mi",null,"∂"),t("msup",null,[t("mi",null,"u"),t("mi",null,"β")])]),t("mrow",null,[t("mi",null,"∂"),t("msup",null,[t("mi",null,"x"),t("mi",null,"α")])])]),t("mo",null,".")])],-1))]),s[43]||(s[43]=t("p",null,"Differentiable version.",-1)),s[44]||(s[44]=t("p",null,[t("a",{href:"https://github.com/agdestein/IncompressibleNavierStokes.jl/blob/a63d20e3a410f14ba6cc612a27e89f8974019d9f/src/operators.jl#L1425",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",E,[t("summary",null,[s[45]||(s[45]=t("a",{id:"IncompressibleNavierStokes.applybodyforce!-NTuple{4, Any}",href:"#IncompressibleNavierStokes.applybodyforce!-NTuple{4, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.applybodyforce!")],-1)),s[46]||(s[46]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[47]||(s[47]=a('
julia
applybodyforce!(F, u, t, setup) -> Any
Compute body force (in-place version). Add the result to F.
source
',3))]),t("details",M,[t("summary",null,[s[48]||(s[48]=t("a",{id:"IncompressibleNavierStokes.applybodyforce-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.applybodyforce-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.applybodyforce")],-1)),s[49]||(s[49]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[50]||(s[50]=a('
julia
applybodyforce(u, t, setup) -> Any
Compute body force (differentiable version).
source
',3))]),t("details",A,[t("summary",null,[s[51]||(s[51]=t("a",{id:"IncompressibleNavierStokes.applypressure!-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.applypressure!-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.applypressure!")],-1)),s[52]||(s[52]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[53]||(s[53]=a('
julia
applypressure!(u, p, setup) -> Any
Subtract pressure gradient (in-place version).
source
',3))]),t("details",C,[t("summary",null,[s[54]||(s[54]=t("a",{id:"IncompressibleNavierStokes.applypressure-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.applypressure-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.applypressure")],-1)),s[55]||(s[55]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[56]||(s[56]=a('
julia
applypressure(u, p, setup) -> Any
Subtract pressure gradient (differentiable version).
source
',3))]),t("details",V,[t("summary",null,[s[57]||(s[57]=t("a",{id:"IncompressibleNavierStokes.avg-NTuple{4, Any}",href:"#IncompressibleNavierStokes.avg-NTuple{4, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.avg")],-1)),s[58]||(s[58]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[59]||(s[59]=a('
julia
avg(ϕ, Δ, I, α) -> Any
Average scalar field ϕ in the α-direction.
source
',3))]),t("details",Z,[t("summary",null,[s[60]||(s[60]=t("a",{id:"IncompressibleNavierStokes.convection!-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.convection!-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.convection!")],-1)),s[61]||(s[61]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[62]||(s[62]=a('
julia
convection!(F, u, setup) -> Any
Compute convective term (in-place version). Add the result to F.
source
',3))]),t("details",S,[t("summary",null,[s[63]||(s[63]=t("a",{id:"IncompressibleNavierStokes.convection-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.convection-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.convection")],-1)),s[64]||(s[64]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[65]||(s[65]=a('
julia
convection(u, setup) -> Any
Compute convective term (differentiable version).
source
',3))]),t("details",N,[t("summary",null,[s[66]||(s[66]=t("a",{id:"IncompressibleNavierStokes.convection_diffusion_temp!-NTuple{4, Any}",href:"#IncompressibleNavierStokes.convection_diffusion_temp!-NTuple{4, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.convection_diffusion_temp!")],-1)),s[67]||(s[67]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[68]||(s[68]=a('
julia
convection_diffusion_temp!(c, u, temp, setup) -> Any
Compute convection-diffusion term for the temperature equation. (in-place version). Add result to c.
source
',3))]),t("details",I,[t("summary",null,[s[69]||(s[69]=t("a",{id:"IncompressibleNavierStokes.convection_diffusion_temp-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.convection_diffusion_temp-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.convection_diffusion_temp")],-1)),s[70]||(s[70]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[71]||(s[71]=a('
julia
convection_diffusion_temp(u, temp, setup) -> Any
Compute convection-diffusion term for the temperature equation. (differentiable version).
source
',3))]),t("details",D,[t("summary",null,[s[72]||(s[72]=t("a",{id:"IncompressibleNavierStokes.convectiondiffusion!-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.convectiondiffusion!-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.convectiondiffusion!")],-1)),s[73]||(s[73]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[74]||(s[74]=a('
julia
convectiondiffusion!(F, u, setup) -> Any
Compute convective and diffusive terms (in-place version). Add the result to F.
source
',3))]),t("details",F,[t("summary",null,[s[75]||(s[75]=t("a",{id:"IncompressibleNavierStokes.diffusion!-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.diffusion!-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.diffusion!")],-1)),s[76]||(s[76]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[77]||(s[77]=a('
julia
diffusion!(F, u, setup; use_viscosity) -> Any
Compute diffusive term (in-place version). Add the result to F.
Keyword arguments:
  • with_viscosity = true: Include viscosity in the operator.
source
',5))]),t("details",O,[t("summary",null,[s[78]||(s[78]=t("a",{id:"IncompressibleNavierStokes.diffusion-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.diffusion-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.diffusion")],-1)),s[79]||(s[79]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[80]||(s[80]=a('
julia
diffusion(u, setup; kwargs...) -> Any
Compute diffusive term (differentiable version).
source
',3))]),t("details",B,[t("summary",null,[s[81]||(s[81]=t("a",{id:"IncompressibleNavierStokes.dissipation!-NTuple{4, Any}",href:"#IncompressibleNavierStokes.dissipation!-NTuple{4, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.dissipation!")],-1)),s[82]||(s[82]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[83]||(s[83]=a('
julia
dissipation!(diss, diff, u, setup) -> Any
Compute dissipation term for the temperature equation (in-place version). Add result to diss.
source
',3))]),t("details",R,[t("summary",null,[s[84]||(s[84]=t("a",{id:"IncompressibleNavierStokes.dissipation-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.dissipation-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.dissipation")],-1)),s[85]||(s[85]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[86]||(s[86]=a('
julia
dissipation(u, setup) -> Any
Compute dissipation term for the temperature equation (differentiable version).
source
',3))]),t("details",G,[t("summary",null,[s[87]||(s[87]=t("a",{id:"IncompressibleNavierStokes.dissipation_from_strain!-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.dissipation_from_strain!-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.dissipation_from_strain!")],-1)),s[88]||(s[88]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[89]||(s[89]=a('
julia
dissipation_from_strain!(ϵ, u, setup) -> Any
Compute dissipation term from strain-rate tensor (in-place version).
source
',3))]),t("details",J,[t("summary",null,[s[90]||(s[90]=t("a",{id:"IncompressibleNavierStokes.dissipation_from_strain-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.dissipation_from_strain-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.dissipation_from_strain")],-1)),s[91]||(s[91]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[96]||(s[96]=a('
julia
dissipation_from_strain(u, setup) -> Any
',1)),t("p",null,[s[94]||(s[94]=e("Compute dissipation term ")),t("mjx-container",z,[(o(),n("svg",X,s[92]||(s[92]=[a('',1)]))),s[93]||(s[93]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mn",null,"2"),t("mi",null,"ν"),t("mo",{fence:"false",stretchy:"false"},"⟨"),t("msub",null,[t("mi",null,"S"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",null,"i"),t("mi",null,"j")])]),t("msub",null,[t("mi",null,"S"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",null,"i"),t("mi",null,"j")])]),t("mo",{fence:"false",stretchy:"false"},"⟩")])],-1))]),s[95]||(s[95]=e(" from strain-rate tensor (differentiable version)."))]),s[97]||(s[97]=t("p",null,[t("a",{href:"https://github.com/agdestein/IncompressibleNavierStokes.jl/blob/a63d20e3a410f14ba6cc612a27e89f8974019d9f/src/operators.jl#L810",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",q,[t("summary",null,[s[98]||(s[98]=t("a",{id:"IncompressibleNavierStokes.divergence!-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.divergence!-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.divergence!")],-1)),s[99]||(s[99]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[100]||(s[100]=a('
julia
divergence!(div, u, setup) -> Any
Compute divergence of velocity field (in-place version).
source
',3))]),t("details",P,[t("summary",null,[s[101]||(s[101]=t("a",{id:"IncompressibleNavierStokes.divergence-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.divergence-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.divergence")],-1)),s[102]||(s[102]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[103]||(s[103]=a('
julia
divergence(u, setup) -> Any
Compute divergence of velocity field (differentiable version).
source
',3))]),t("details",K,[t("summary",null,[s[104]||(s[104]=t("a",{id:"IncompressibleNavierStokes.divoftensor!-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.divoftensor!-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.divoftensor!")],-1)),s[105]||(s[105]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[106]||(s[106]=a('
julia
divoftensor!(s, σ, setup) -> Any
Compute divergence of a tensor with all components in the pressure points (in-place version). The stress tensors should be precomputed and stored in σ.
source
',3))]),t("details",U,[t("summary",null,[s[107]||(s[107]=t("a",{id:"IncompressibleNavierStokes.eig2field!-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.eig2field!-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.eig2field!")],-1)),s[108]||(s[108]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[113]||(s[113]=a('
julia
eig2field!(λ, u, setup) -> Any
',1)),t("p",null,[s[111]||(s[111]=e("Compute the second eigenvalue of ")),t("mjx-container",$,[(o(),n("svg",W,s[109]||(s[109]=[a('',1)]))),s[110]||(s[110]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msup",null,[t("mi",null,"S"),t("mn",null,"2")]),t("mo",null,"+"),t("msup",null,[t("mi",null,"R"),t("mn",null,"2")])])],-1))]),s[112]||(s[112]=e(" (in-place version)."))]),s[114]||(s[114]=t("p",null,[t("a",{href:"https://github.com/agdestein/IncompressibleNavierStokes.jl/blob/a63d20e3a410f14ba6cc612a27e89f8974019d9f/src/operators.jl#L1470",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",Y,[t("summary",null,[s[115]||(s[115]=t("a",{id:"IncompressibleNavierStokes.eig2field-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.eig2field-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.eig2field")],-1)),s[116]||(s[116]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[123]||(s[123]=a('
julia
eig2field(u, setup) -> Any
',1)),t("p",null,[s[119]||(s[119]=e("Compute the second eigenvalue of ")),t("mjx-container",_,[(o(),n("svg",t1,s[117]||(s[117]=[a('',1)]))),s[118]||(s[118]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msup",null,[t("mi",null,"S"),t("mn",null,"2")]),t("mo",null,"+"),t("msup",null,[t("mi",null,"R"),t("mn",null,"2")])])],-1))]),s[120]||(s[120]=e(", as proposed by Jeong and Hussain [")),s[121]||(s[121]=t("a",{href:"/IncompressibleNavierStokes.jl/dev/references#Jeong1995"},"6",-1)),s[122]||(s[122]=e("]."))]),s[124]||(s[124]=t("p",null,"Differentiable version.",-1)),s[125]||(s[125]=t("p",null,[t("a",{href:"https://github.com/agdestein/IncompressibleNavierStokes.jl/blob/a63d20e3a410f14ba6cc612a27e89f8974019d9f/src/operators.jl#L1462",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",s1,[t("summary",null,[s[126]||(s[126]=t("a",{id:"IncompressibleNavierStokes.get_scale_numbers-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.get_scale_numbers-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.get_scale_numbers")],-1)),s[127]||(s[127]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[149]||(s[149]=a(`
julia
get_scale_numbers(
     u,
     setup
-) -> NamedTuple{(:uavg, :ϵ, :η, :λ, :Reλ, :L, :τ), <:Tuple{Any, Any, Any, Any, Any, Nothing, Nothing}}
Get the following dimensional scale numbers [7]:
`,2)),t("ul",null,[t("li",null,[t("p",null,[s[130]||(s[130]=e("Velocity ")),t("mjx-container",e1,[(o(),n("svg",a1,s[128]||(s[128]=[a('',1)]))),s[129]||(s[129]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"u"),t("mtext",null,"avg")]),t("mo",null,"="),t("mo",{fence:"false",stretchy:"false"},"⟨"),t("msub",null,[t("mi",null,"u"),t("mi",null,"i")]),t("msub",null,[t("mi",null,"u"),t("mi",null,"i")]),t("msup",null,[t("mo",{fence:"false",stretchy:"false"},"⟩"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mn",null,"1"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mo",null,"/")]),t("mn",null,"2")])])])],-1))])])]),t("li",null,[t("p",null,[s[133]||(s[133]=e("Dissipation rate ")),t("mjx-container",l1,[(o(),n("svg",i1,s[131]||(s[131]=[a('',1)]))),s[132]||(s[132]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 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julia
gravity!(F, temp, setup) -> Any
Compute gravity term (in-place version). add the result to F.
source
',3))]),t("details",k1,[t("summary",null,[s[154]||(s[154]=t("a",{id:"IncompressibleNavierStokes.gravity-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.gravity-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.gravity")],-1)),s[155]||(s[155]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[156]||(s[156]=a('
julia
gravity(temp, setup) -> Any
Compute gravity term (differentiable version).
source
',3))]),t("details",c1,[t("summary",null,[s[157]||(s[157]=t("a",{id:"IncompressibleNavierStokes.interpolate_u_p!-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.interpolate_u_p!-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.interpolate_u_p!")],-1)),s[158]||(s[158]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[159]||(s[159]=a('
julia
interpolate_u_p!(up, u, setup) -> Any
Interpolate velocity to pressure points (in-place version).
source
',3))]),t("details",y1,[t("summary",null,[s[160]||(s[160]=t("a",{id:"IncompressibleNavierStokes.interpolate_u_p-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.interpolate_u_p-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.interpolate_u_p")],-1)),s[161]||(s[161]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[162]||(s[162]=a('
julia
interpolate_u_p(u, setup) -> Any
Interpolate velocity to pressure points (differentiable version).
source
',3))]),t("details",b1,[t("summary",null,[s[163]||(s[163]=t("a",{id:"IncompressibleNavierStokes.interpolate_ω_p!-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.interpolate_ω_p!-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.interpolate_ω_p!")],-1)),s[164]||(s[164]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[165]||(s[165]=a('
julia
interpolate_ω_p!(ωp, ω, setup) -> Any
Interpolate vorticity to pressure points (in-place version).
source
',3))]),t("details",f1,[t("summary",null,[s[166]||(s[166]=t("a",{id:"IncompressibleNavierStokes.interpolate_ω_p-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.interpolate_ω_p-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.interpolate_ω_p")],-1)),s[167]||(s[167]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[168]||(s[168]=a('
julia
interpolate_ω_p(ω, setup) -> Any
Interpolate vorticity to pressure points (differentiable version).
source
',3))]),t("details",v1,[t("summary",null,[s[169]||(s[169]=t("a",{id:"IncompressibleNavierStokes.kinetic_energy!-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.kinetic_energy!-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.kinetic_energy!")],-1)),s[170]||(s[170]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[171]||(s[171]=a('
julia
kinetic_energy!(ke, u, setup; interpolate_first) -> Any
Compute kinetic energy field (in-place version).
source
',3))]),t("details",x1,[t("summary",null,[s[172]||(s[172]=t("a",{id:"IncompressibleNavierStokes.kinetic_energy-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.kinetic_energy-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.kinetic_energy")],-1)),s[173]||(s[173]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[184]||(s[184]=a('
julia
kinetic_energy(u, setup; kwargs...) -> Any
',1)),t("p",null,[s[176]||(s[176]=e("Compute kinetic energy field ")),t("mjx-container",w1,[(o(),n("svg",j1,s[174]||(s[174]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D458",d:"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z",style:{"stroke-width":"3"}})])])],-1)]))),s[175]||(s[175]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"k")])],-1))]),s[177]||(s[177]=e(" (in-place version). 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julia
laplacian!(L, p, setup) -> Any
Compute Laplacian of pressure field (in-place version).
source
',3))]),t("details",C1,[t("summary",null,[s[192]||(s[192]=t("a",{id:"IncompressibleNavierStokes.laplacian-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.laplacian-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.laplacian")],-1)),s[193]||(s[193]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[194]||(s[194]=a('
julia
laplacian(p, setup) -> Any
Compute Laplacian of pressure field (differentiable version).
source
',3))]),t("details",V1,[t("summary",null,[s[195]||(s[195]=t("a",{id:"IncompressibleNavierStokes.momentum!-NTuple{5, Any}",href:"#IncompressibleNavierStokes.momentum!-NTuple{5, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.momentum!")],-1)),s[196]||(s[196]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[197]||(s[197]=a('
julia
momentum!(F, u, temp, t, setup) -> Any
Right hand side of momentum equations, excluding pressure gradient (in-place version).
source
',3))]),t("details",Z1,[t("summary",null,[s[198]||(s[198]=t("a",{id:"IncompressibleNavierStokes.momentum-NTuple{4, Any}",href:"#IncompressibleNavierStokes.momentum-NTuple{4, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.momentum")],-1)),s[199]||(s[199]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[200]||(s[200]=a('
julia
momentum(u, temp, t, setup) -> Any
Right hand side of momentum equations, excluding pressure gradient (differentiable version).
source
',3))]),t("details",S1,[t("summary",null,[s[201]||(s[201]=t("a",{id:"IncompressibleNavierStokes.pressuregradient!-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.pressuregradient!-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.pressuregradient!")],-1)),s[202]||(s[202]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[203]||(s[203]=a('
julia
pressuregradient!(G, p, setup) -> Any
Compute pressure gradient (in-place version).
source
',3))]),t("details",N1,[t("summary",null,[s[204]||(s[204]=t("a",{id:"IncompressibleNavierStokes.pressuregradient-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.pressuregradient-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.pressuregradient")],-1)),s[205]||(s[205]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[206]||(s[206]=a('
julia
pressuregradient(p, setup) -> Any
Compute pressure gradient (differentiable version).
source
',3))]),t("details",I1,[t("summary",null,[s[207]||(s[207]=t("a",{id:"IncompressibleNavierStokes.scalewithvolume!-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.scalewithvolume!-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.scalewithvolume!")],-1)),s[208]||(s[208]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[209]||(s[209]=a('
julia
scalewithvolume!(p, setup) -> Any
Scale scalar field with volume sizes (in-place version).
source
',3))]),t("details",D1,[t("summary",null,[s[210]||(s[210]=t("a",{id:"IncompressibleNavierStokes.scalewithvolume-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.scalewithvolume-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.scalewithvolume")],-1)),s[211]||(s[211]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[212]||(s[212]=a('
julia
scalewithvolume(p, setup) -> Any
Scale scalar field p with volume sizes (differentiable version).
source
',3))]),t("details",F1,[t("summary",null,[s[213]||(s[213]=t("a",{id:"IncompressibleNavierStokes.smagorinsky_closure-Tuple{Any}",href:"#IncompressibleNavierStokes.smagorinsky_closure-Tuple{Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.smagorinsky_closure")],-1)),s[214]||(s[214]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[215]||(s[215]=a(`
julia
smagorinsky_closure(
+) -> NamedTuple{(:uavg, :ϵ, :η, :λ, :Reλ, :L, :τ), <:Tuple{Any, Any, Any, Any, Any, Nothing, Nothing}}
Get the following dimensional scale numbers [7]:
`,2)),t("ul",null,[t("li",null,[t("p",null,[s[130]||(s[130]=e("Velocity ")),t("mjx-container",e1,[(o(),n("svg",a1,s[128]||(s[128]=[a('',1)]))),s[129]||(s[129]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"u"),t("mtext",null,"avg")]),t("mo",null,"="),t("mo",{fence:"false",stretchy:"false"},"⟨"),t("msub",null,[t("mi",null,"u"),t("mi",null,"i")]),t("msub",null,[t("mi",null,"u"),t("mi",null,"i")]),t("msup",null,[t("mo",{fence:"false",stretchy:"false"},"⟩"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mn",null,"1"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mo",null,"/")]),t("mn",null,"2")])])])],-1))])])]),t("li",null,[t("p",null,[s[133]||(s[133]=e("Dissipation rate ")),t("mjx-container",l1,[(o(),n("svg",i1,s[131]||(s[131]=[a('',1)]))),s[132]||(s[132]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 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")),t("mjx-container",h1,[(o(),n("svg",g1,s[146]||(s[146]=[a('',1)]))),s[147]||(s[147]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"τ"),t("mo",null,"="),t("mfrac",null,[t("mi",null,"L"),t("msub",null,[t("mi",null,"u"),t("mtext",null,"avg")])])])],-1))])])])]),s[150]||(s[150]=t("p",null,[t("a",{href:"https://github.com/agdestein/IncompressibleNavierStokes.jl/blob/a63d20e3a410f14ba6cc612a27e89f8974019d9f/src/operators.jl#L1558",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",u1,[t("summary",null,[s[151]||(s[151]=t("a",{id:"IncompressibleNavierStokes.gravity!-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.gravity!-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.gravity!")],-1)),s[152]||(s[152]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[153]||(s[153]=a('
julia
gravity!(F, temp, setup) -> Any
Compute gravity term (in-place version). add the result to F.
source
',3))]),t("details",k1,[t("summary",null,[s[154]||(s[154]=t("a",{id:"IncompressibleNavierStokes.gravity-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.gravity-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.gravity")],-1)),s[155]||(s[155]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[156]||(s[156]=a('
julia
gravity(temp, setup) -> Any
Compute gravity term (differentiable version).
source
',3))]),t("details",c1,[t("summary",null,[s[157]||(s[157]=t("a",{id:"IncompressibleNavierStokes.interpolate_u_p!-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.interpolate_u_p!-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.interpolate_u_p!")],-1)),s[158]||(s[158]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[159]||(s[159]=a('
julia
interpolate_u_p!(up, u, setup) -> Any
Interpolate velocity to pressure points (in-place version).
source
',3))]),t("details",y1,[t("summary",null,[s[160]||(s[160]=t("a",{id:"IncompressibleNavierStokes.interpolate_u_p-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.interpolate_u_p-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.interpolate_u_p")],-1)),s[161]||(s[161]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[162]||(s[162]=a('
julia
interpolate_u_p(u, setup) -> Any
Interpolate velocity to pressure points (differentiable version).
source
',3))]),t("details",b1,[t("summary",null,[s[163]||(s[163]=t("a",{id:"IncompressibleNavierStokes.interpolate_ω_p!-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.interpolate_ω_p!-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.interpolate_ω_p!")],-1)),s[164]||(s[164]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[165]||(s[165]=a('
julia
interpolate_ω_p!(ωp, ω, setup) -> Any
Interpolate vorticity to pressure points (in-place version).
source
',3))]),t("details",f1,[t("summary",null,[s[166]||(s[166]=t("a",{id:"IncompressibleNavierStokes.interpolate_ω_p-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.interpolate_ω_p-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.interpolate_ω_p")],-1)),s[167]||(s[167]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[168]||(s[168]=a('
julia
interpolate_ω_p(ω, setup) -> Any
Interpolate vorticity to pressure points (differentiable version).
source
',3))]),t("details",v1,[t("summary",null,[s[169]||(s[169]=t("a",{id:"IncompressibleNavierStokes.kinetic_energy!-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.kinetic_energy!-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.kinetic_energy!")],-1)),s[170]||(s[170]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[171]||(s[171]=a('
julia
kinetic_energy!(ke, u, setup; interpolate_first) -> Any
Compute kinetic energy field (in-place version).
source
',3))]),t("details",x1,[t("summary",null,[s[172]||(s[172]=t("a",{id:"IncompressibleNavierStokes.kinetic_energy-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.kinetic_energy-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.kinetic_energy")],-1)),s[173]||(s[173]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[184]||(s[184]=a('
julia
kinetic_energy(u, setup; kwargs...) -> Any
',1)),t("p",null,[s[176]||(s[176]=e("Compute kinetic energy field ")),t("mjx-container",w1,[(o(),n("svg",j1,s[174]||(s[174]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D458",d:"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z",style:{"stroke-width":"3"}})])])],-1)]))),s[175]||(s[175]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"k")])],-1))]),s[177]||(s[177]=e(" (in-place version). If ")),s[178]||(s[178]=t("code",null,"interpolate_first",-1)),s[179]||(s[179]=e(" is true, it is given by"))]),t("mjx-container",H1,[(o(),n("svg",L1,s[180]||(s[180]=[a('',1)]))),s[181]||(s[181]=t("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[t("msub",null,[t("mi",null,"k"),t("mi",null,"I")]),t("mo",null,"="),t("mfrac",null,[t("mn",null,"1"),t("mn",null,"8")]),t("munder",null,[t("mo",{"data-mjx-texclass":"OP"},"∑"),t("mi",null,"α")]),t("mo",{stretchy:"false"},"("),t("msubsup",null,[t("mi",null,"u"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",null,"I"),t("mo",null,"+"),t("msub",null,[t("mi",null,"h"),t("mi",null,"α")])]),t("mi",null,"α")]),t("mo",null,"+"),t("msubsup",null,[t("mi",null,"u"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",null,"I"),t("mo",null,"−"),t("msub",null,[t("mi",null,"h"),t("mi",null,"α")])]),t("mi",null,"α")]),t("msup",null,[t("mo",{stretchy:"false"},")"),t("mn",null,"2")]),t("mo",null,".")])],-1))]),s[185]||(s[185]=t("p",null,"Otherwise, it is given by",-1)),t("mjx-container",E1,[(o(),n("svg",M1,s[182]||(s[182]=[a('',1)]))),s[183]||(s[183]=t("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[t("msub",null,[t("mi",null,"k"),t("mi",null,"I")]),t("mo",null,"="),t("mfrac",null,[t("mn",null,"1"),t("mn",null,"4")]),t("munder",null,[t("mo",{"data-mjx-texclass":"OP"},"∑"),t("mi",null,"α")]),t("mo",{stretchy:"false"},"("),t("mo",{stretchy:"false"},"("),t("msubsup",null,[t("mi",null,"u"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",null,"I"),t("mo",null,"+"),t("msub",null,[t("mi",null,"h"),t("mi",null,"α")])]),t("mi",null,"α")]),t("msup",null,[t("mo",{stretchy:"false"},")"),t("mn",null,"2")]),t("mo",null,"+"),t("mo",{stretchy:"false"},"("),t("msubsup",null,[t("mi",null,"u"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",null,"I"),t("mo",null,"−"),t("msub",null,[t("mi",null,"h"),t("mi",null,"α")])]),t("mi",null,"α")]),t("msup",null,[t("mo",{stretchy:"false"},")"),t("mn",null,"2")]),t("mo",{stretchy:"false"},")"),t("mo",null,",")])],-1))]),s[186]||(s[186]=t("p",null,[e("as in ["),t("a",{href:"/IncompressibleNavierStokes.jl/dev/references#Sanderse2023"},"8"),e("].")],-1)),s[187]||(s[187]=t("p",null,"Differentiable version.",-1)),s[188]||(s[188]=t("p",null,[t("a",{href:"https://github.com/agdestein/IncompressibleNavierStokes.jl/blob/a63d20e3a410f14ba6cc612a27e89f8974019d9f/src/operators.jl#L1491",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",A1,[t("summary",null,[s[189]||(s[189]=t("a",{id:"IncompressibleNavierStokes.laplacian!-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.laplacian!-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.laplacian!")],-1)),s[190]||(s[190]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[191]||(s[191]=a('
julia
laplacian!(L, p, setup) -> Any
Compute Laplacian of pressure field (in-place version).
source
',3))]),t("details",C1,[t("summary",null,[s[192]||(s[192]=t("a",{id:"IncompressibleNavierStokes.laplacian-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.laplacian-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.laplacian")],-1)),s[193]||(s[193]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[194]||(s[194]=a('
julia
laplacian(p, setup) -> Any
Compute Laplacian of pressure field (differentiable version).
source
',3))]),t("details",V1,[t("summary",null,[s[195]||(s[195]=t("a",{id:"IncompressibleNavierStokes.momentum!-NTuple{5, Any}",href:"#IncompressibleNavierStokes.momentum!-NTuple{5, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.momentum!")],-1)),s[196]||(s[196]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[197]||(s[197]=a('
julia
momentum!(F, u, temp, t, setup) -> Any
Right hand side of momentum equations, excluding pressure gradient (in-place version).
source
',3))]),t("details",Z1,[t("summary",null,[s[198]||(s[198]=t("a",{id:"IncompressibleNavierStokes.momentum-NTuple{4, Any}",href:"#IncompressibleNavierStokes.momentum-NTuple{4, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.momentum")],-1)),s[199]||(s[199]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[200]||(s[200]=a('
julia
momentum(u, temp, t, setup) -> Any
Right hand side of momentum equations, excluding pressure gradient (differentiable version).
source
',3))]),t("details",S1,[t("summary",null,[s[201]||(s[201]=t("a",{id:"IncompressibleNavierStokes.pressuregradient!-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.pressuregradient!-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.pressuregradient!")],-1)),s[202]||(s[202]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[203]||(s[203]=a('
julia
pressuregradient!(G, p, setup) -> Any
Compute pressure gradient (in-place version).
source
',3))]),t("details",N1,[t("summary",null,[s[204]||(s[204]=t("a",{id:"IncompressibleNavierStokes.pressuregradient-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.pressuregradient-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.pressuregradient")],-1)),s[205]||(s[205]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[206]||(s[206]=a('
julia
pressuregradient(p, setup) -> Any
Compute pressure gradient (differentiable version).
source
',3))]),t("details",I1,[t("summary",null,[s[207]||(s[207]=t("a",{id:"IncompressibleNavierStokes.scalewithvolume!-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.scalewithvolume!-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.scalewithvolume!")],-1)),s[208]||(s[208]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[209]||(s[209]=a('
julia
scalewithvolume!(p, setup) -> Any
Scale scalar field with volume sizes (in-place version).
source
',3))]),t("details",D1,[t("summary",null,[s[210]||(s[210]=t("a",{id:"IncompressibleNavierStokes.scalewithvolume-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.scalewithvolume-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.scalewithvolume")],-1)),s[211]||(s[211]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[212]||(s[212]=a('
julia
scalewithvolume(p, setup) -> Any
Scale scalar field p with volume sizes (differentiable version).
source
',3))]),t("details",F1,[t("summary",null,[s[213]||(s[213]=t("a",{id:"IncompressibleNavierStokes.smagorinsky_closure-Tuple{Any}",href:"#IncompressibleNavierStokes.smagorinsky_closure-Tuple{Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.smagorinsky_closure")],-1)),s[214]||(s[214]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[215]||(s[215]=a(`
julia
smagorinsky_closure(
     setup
-) -> IncompressibleNavierStokes.var"#closure#188"
Create Smagorinsky closure model m. The model is called as m(u, θ), where the Smagorinsky constant θ should be a scalar between 0 and 1 (for example θ = 0.1).
source
`,3))]),t("details",O1,[t("summary",null,[s[216]||(s[216]=t("a",{id:"IncompressibleNavierStokes.smagtensor!-NTuple{4, Any}",href:"#IncompressibleNavierStokes.smagtensor!-NTuple{4, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.smagtensor!")],-1)),s[217]||(s[217]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[218]||(s[218]=a('
julia
smagtensor!(σ, u, θ, setup) -> Any
Compute Smagorinsky stress tensors σ[I] (in-place version). The Smagorinsky constant θ should be a scalar between 0 and 1.
source
',3))]),t("details",B1,[t("summary",null,[s[219]||(s[219]=t("a",{id:"IncompressibleNavierStokes.total_kinetic_energy-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.total_kinetic_energy-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.total_kinetic_energy")],-1)),s[220]||(s[220]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[221]||(s[221]=a('
julia
total_kinetic_energy(u, setup; kwargs...) -> Any
Compute total kinetic energy. The velocity components are interpolated to the volume centers and squared.
source
',3))]),t("details",R1,[t("summary",null,[s[222]||(s[222]=t("a",{id:"IncompressibleNavierStokes.unit_cartesian_indices-Tuple{Any}",href:"#IncompressibleNavierStokes.unit_cartesian_indices-Tuple{Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.unit_cartesian_indices")],-1)),s[223]||(s[223]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[224]||(s[224]=a('
julia
unit_cartesian_indices(D) -> Any
Get tuple of all unit vectors as Cartesian indices.
source
',3))]),t("details",G1,[t("summary",null,[s[225]||(s[225]=t("a",{id:"IncompressibleNavierStokes.vorticity!-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.vorticity!-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.vorticity!")],-1)),s[226]||(s[226]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[227]||(s[227]=a('
julia
vorticity!(ω, u, setup) -> Any
Compute vorticity field (in-place version).
source
',3))]),t("details",J1,[t("summary",null,[s[228]||(s[228]=t("a",{id:"IncompressibleNavierStokes.vorticity-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.vorticity-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.vorticity")],-1)),s[229]||(s[229]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[230]||(s[230]=a('
julia
vorticity(u, setup) -> Any
Compute vorticity field (differentiable version).
source
',3))]),t("details",z1,[t("summary",null,[s[231]||(s[231]=t("a",{id:"IncompressibleNavierStokes.lastdimcontract-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.lastdimcontract-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.lastdimcontract")],-1)),s[232]||(s[232]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[233]||(s[233]=a('
julia
lastdimcontract(a, b, setup) -> Any
Compute c[I] = sum_i a[I, i] * b[I, i], where c[:, i] and b[:, i] are tensor fields (elements are SMatrixes), a[:, i] is a scalar field, and the i dimension is contracted (multiple channels).
source
',3))]),t("details",X1,[t("summary",null,[s[234]||(s[234]=t("a",{id:"IncompressibleNavierStokes.tensorbasis!-NTuple{4, Any}",href:"#IncompressibleNavierStokes.tensorbasis!-NTuple{4, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.tensorbasis!")],-1)),s[235]||(s[235]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[257]||(s[257]=a('
julia
tensorbasis!(B, V, u, setup) -> Tuple{Any, Any}
',1)),t("p",null,[s[240]||(s[240]=e("Compute symmetry tensor basis ")),s[241]||(s[241]=t("code",null,"B[1]",-1)),s[242]||(s[242]=e("-")),s[243]||(s[243]=t("code",null,"B[11]",-1)),s[244]||(s[244]=e(" and invariants ")),s[245]||(s[245]=t("code",null,"V[1]",-1)),s[246]||(s[246]=e("-")),s[247]||(s[247]=t("code",null,"V[5]",-1)),s[248]||(s[248]=e(", as specified in [")),s[249]||(s[249]=t("a",{href:"/IncompressibleNavierStokes.jl/dev/references#Silvis2017"},"9",-1)),s[250]||(s[250]=e("] in equations (9) and (11). Note that ")),s[251]||(s[251]=t("code",null,"B[1]",-1)),s[252]||(s[252]=e(" corresponds to ")),t("mjx-container",q1,[(o(),n("svg",P1,s[236]||(s[236]=[a('',1)]))),s[237]||(s[237]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"T"),t("mn",null,"0")])])],-1))]),s[253]||(s[253]=e(" in the paper, and ")),s[254]||(s[254]=t("code",null,"V",-1)),s[255]||(s[255]=e(" to ")),t("mjx-container",K1,[(o(),n("svg",U1,s[238]||(s[238]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D43C",d:"M43 1Q26 1 26 10Q26 12 29 24Q34 43 39 45Q42 46 54 46H60Q120 46 136 53Q137 53 138 54Q143 56 149 77T198 273Q210 318 216 344Q286 624 286 626Q284 630 284 631Q274 637 213 637H193Q184 643 189 662Q193 677 195 680T209 683H213Q285 681 359 681Q481 681 487 683H497Q504 676 504 672T501 655T494 639Q491 637 471 637Q440 637 407 634Q393 631 388 623Q381 609 337 432Q326 385 315 341Q245 65 245 59Q245 52 255 50T307 46H339Q345 38 345 37T342 19Q338 6 332 0H316Q279 2 179 2Q143 2 113 2T65 2T43 1Z",style:{"stroke-width":"3"}})])])],-1)]))),s[239]||(s[239]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"I")])],-1))]),s[256]||(s[256]=e("."))]),s[258]||(s[258]=t("p",null,[t("a",{href:"https://github.com/agdestein/IncompressibleNavierStokes.jl/blob/03acafa3566742b1f37d722d2207f5dc21d031c0/src/tensorbasis.jl#L16",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",$1,[t("summary",null,[s[259]||(s[259]=t("a",{id:"IncompressibleNavierStokes.tensorbasis-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.tensorbasis-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.tensorbasis")],-1)),s[260]||(s[260]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[261]||(s[261]=a('
julia
tensorbasis(u, setup) -> Tuple{Any, Any}
Compute symmetry tensor basis (differentiable version).
source
',3))])])}const i2=Q(T,[["render",W1]]);export{l2 as __pageData,i2 as default};
+) -> IncompressibleNavierStokes.var"#closure#188"
Create Smagorinsky closure model m. The model is called as m(u, θ), where the Smagorinsky constant θ should be a scalar between 0 and 1 (for example θ = 0.1).
source
`,3))]),t("details",O1,[t("summary",null,[s[216]||(s[216]=t("a",{id:"IncompressibleNavierStokes.smagtensor!-NTuple{4, Any}",href:"#IncompressibleNavierStokes.smagtensor!-NTuple{4, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.smagtensor!")],-1)),s[217]||(s[217]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[218]||(s[218]=a('
julia
smagtensor!(σ, u, θ, setup) -> Any
Compute Smagorinsky stress tensors σ[I] (in-place version). The Smagorinsky constant θ should be a scalar between 0 and 1.
source
',3))]),t("details",B1,[t("summary",null,[s[219]||(s[219]=t("a",{id:"IncompressibleNavierStokes.total_kinetic_energy-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.total_kinetic_energy-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.total_kinetic_energy")],-1)),s[220]||(s[220]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[221]||(s[221]=a('
julia
total_kinetic_energy(u, setup; kwargs...) -> Any
Compute total kinetic energy. The velocity components are interpolated to the volume centers and squared.
source
',3))]),t("details",R1,[t("summary",null,[s[222]||(s[222]=t("a",{id:"IncompressibleNavierStokes.unit_cartesian_indices-Tuple{Any}",href:"#IncompressibleNavierStokes.unit_cartesian_indices-Tuple{Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.unit_cartesian_indices")],-1)),s[223]||(s[223]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[224]||(s[224]=a('
julia
unit_cartesian_indices(D) -> Any
Get tuple of all unit vectors as Cartesian indices.
source
',3))]),t("details",G1,[t("summary",null,[s[225]||(s[225]=t("a",{id:"IncompressibleNavierStokes.vorticity!-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.vorticity!-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.vorticity!")],-1)),s[226]||(s[226]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[227]||(s[227]=a('
julia
vorticity!(ω, u, setup) -> Any
Compute vorticity field (in-place version).
source
',3))]),t("details",J1,[t("summary",null,[s[228]||(s[228]=t("a",{id:"IncompressibleNavierStokes.vorticity-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.vorticity-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.vorticity")],-1)),s[229]||(s[229]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[230]||(s[230]=a('
julia
vorticity(u, setup) -> Any
Compute vorticity field (differentiable version).
source
',3))]),t("details",z1,[t("summary",null,[s[231]||(s[231]=t("a",{id:"IncompressibleNavierStokes.lastdimcontract-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.lastdimcontract-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.lastdimcontract")],-1)),s[232]||(s[232]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[233]||(s[233]=a('
julia
lastdimcontract(a, b, setup) -> Any
Compute c[I] = sum_i a[I, i] * b[I, i], where c[:, i] and b[:, i] are tensor fields (elements are SMatrixes), a[:, i] is a scalar field, and the i dimension is contracted (multiple channels).
source
',3))]),t("details",X1,[t("summary",null,[s[234]||(s[234]=t("a",{id:"IncompressibleNavierStokes.tensorbasis!-NTuple{4, Any}",href:"#IncompressibleNavierStokes.tensorbasis!-NTuple{4, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.tensorbasis!")],-1)),s[235]||(s[235]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[257]||(s[257]=a('
julia
tensorbasis!(B, V, u, setup) -> Tuple{Any, Any}
',1)),t("p",null,[s[240]||(s[240]=e("Compute symmetry tensor basis ")),s[241]||(s[241]=t("code",null,"B[1]",-1)),s[242]||(s[242]=e("-")),s[243]||(s[243]=t("code",null,"B[11]",-1)),s[244]||(s[244]=e(" and invariants ")),s[245]||(s[245]=t("code",null,"V[1]",-1)),s[246]||(s[246]=e("-")),s[247]||(s[247]=t("code",null,"V[5]",-1)),s[248]||(s[248]=e(", as specified in [")),s[249]||(s[249]=t("a",{href:"/IncompressibleNavierStokes.jl/dev/references#Silvis2017"},"9",-1)),s[250]||(s[250]=e("] in equations (9) and (11). Note that ")),s[251]||(s[251]=t("code",null,"B[1]",-1)),s[252]||(s[252]=e(" corresponds to ")),t("mjx-container",q1,[(o(),n("svg",P1,s[236]||(s[236]=[a('',1)]))),s[237]||(s[237]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"T"),t("mn",null,"0")])])],-1))]),s[253]||(s[253]=e(" in the paper, and ")),s[254]||(s[254]=t("code",null,"V",-1)),s[255]||(s[255]=e(" to ")),t("mjx-container",K1,[(o(),n("svg",U1,s[238]||(s[238]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D43C",d:"M43 1Q26 1 26 10Q26 12 29 24Q34 43 39 45Q42 46 54 46H60Q120 46 136 53Q137 53 138 54Q143 56 149 77T198 273Q210 318 216 344Q286 624 286 626Q284 630 284 631Q274 637 213 637H193Q184 643 189 662Q193 677 195 680T209 683H213Q285 681 359 681Q481 681 487 683H497Q504 676 504 672T501 655T494 639Q491 637 471 637Q440 637 407 634Q393 631 388 623Q381 609 337 432Q326 385 315 341Q245 65 245 59Q245 52 255 50T307 46H339Q345 38 345 37T342 19Q338 6 332 0H316Q279 2 179 2Q143 2 113 2T65 2T43 1Z",style:{"stroke-width":"3"}})])])],-1)]))),s[239]||(s[239]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"I")])],-1))]),s[256]||(s[256]=e("."))]),s[258]||(s[258]=t("p",null,[t("a",{href:"https://github.com/agdestein/IncompressibleNavierStokes.jl/blob/a63d20e3a410f14ba6cc612a27e89f8974019d9f/src/tensorbasis.jl#L16",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",$1,[t("summary",null,[s[259]||(s[259]=t("a",{id:"IncompressibleNavierStokes.tensorbasis-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.tensorbasis-Tuple{Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.tensorbasis")],-1)),s[260]||(s[260]=e()),i(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[261]||(s[261]=a('
julia
tensorbasis(u, setup) -> Tuple{Any, Any}
Compute symmetry tensor basis (differentiable version).
source
',3))])])}const i2=Q(T,[["render",W1]]);export{l2 as __pageData,i2 as default};
diff --git a/dev/assets/manual_pressure.md.CRdAlBfk.js b/dev/assets/manual_pressure.md.CGOE2Wqh.js
similarity index 95%
rename from dev/assets/manual_pressure.md.CRdAlBfk.js
rename to dev/assets/manual_pressure.md.CGOE2Wqh.js
index 5e80ac24a..b47c4639e 100644
--- a/dev/assets/manual_pressure.md.CRdAlBfk.js
+++ b/dev/assets/manual_pressure.md.CGOE2Wqh.js
@@ -1,16 +1,16 @@
 import{_ as r,c as n,j as e,a as i,a5 as a,G as l,B as o,o as p}from"./chunks/framework.Dr1kDDE8.js";const I=JSON.parse('{"title":"Pressure solvers","description":"","frontmatter":{},"headers":[],"relativePath":"manual/pressure.md","filePath":"manual/pressure.md","lastUpdated":null}'),d={name:"manual/pressure.md"},h={class:"MathJax",jax:"SVG",display:"true",style:{direction:"ltr",display:"block","text-align":"center",margin:"1em 0",position:"relative"}},k={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"15.194ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 6715.6 1000","aria-hidden":"true"},c={class:"jldocstring custom-block",open:""},u={class:"jldocstring custom-block",open:""},g={class:"jldocstring custom-block",open:""},Q={class:"jldocstring custom-block",open:""},m={class:"jldocstring custom-block",open:""},T={class:"jldocstring custom-block",open:""},b={class:"jldocstring custom-block",open:""},y={class:"jldocstring custom-block",open:""},v={class:"jldocstring custom-block",open:""},E={class:"jldocstring custom-block",open:""},f={class:"jldocstring custom-block",open:""};function j(C,s,N,F,S,x){const t=o("Badge");return p(),n("div",null,[s[35]||(s[35]=e("h1",{id:"Pressure-solvers",tabindex:"-1"},[i("Pressure solvers "),e("a",{class:"header-anchor",href:"#Pressure-solvers","aria-label":'Permalink to "Pressure solvers {#Pressure-solvers}"'},"​")],-1)),s[36]||(s[36]=e("p",null,"The discrete pressure Poisson equation",-1)),e("mjx-container",h,[(p(),n("svg",k,s[0]||(s[0]=[a('',1)]))),s[1]||(s[1]=e("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[e("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[e("mi",null,"L"),e("mi",null,"p"),e("mo",null,"="),e("mi",null,"W"),e("mi",null,"M"),e("mi",null,"F"),e("mo",{stretchy:"false"},"("),e("mi",null,"u"),e("mo",{stretchy:"false"},")")])],-1))]),s[37]||(s[37]=e("p",null,"enforces divergence freeness. There are multiple options for solving this system.",-1)),e("details",c,[e("summary",null,[s[2]||(s[2]=e("a",{id:"IncompressibleNavierStokes.default_psolver-Tuple{Any}",href:"#IncompressibleNavierStokes.default_psolver-Tuple{Any}"},[e("span",{class:"jlbinding"},"IncompressibleNavierStokes.default_psolver")],-1)),s[3]||(s[3]=i()),l(t,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[4]||(s[4]=a(`
julia
default_psolver(
     setup
-) -> Union{IncompressibleNavierStokes.var"#psolve!#101"{Bool}, IncompressibleNavierStokes.var"#psolve!#124"}
Get default Poisson solver from setup.
source
`,3))]),e("details",u,[e("summary",null,[s[5]||(s[5]=e("a",{id:"IncompressibleNavierStokes.poisson!-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.poisson!-Tuple{Any, Any}"},[e("span",{class:"jlbinding"},"IncompressibleNavierStokes.poisson!")],-1)),s[6]||(s[6]=i()),l(t,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[7]||(s[7]=a('
julia
poisson!(psolver, f) -> Any
Solve the Poisson equation for the pressure (in-place version).
source
',3))]),e("details",g,[e("summary",null,[s[8]||(s[8]=e("a",{id:"IncompressibleNavierStokes.poisson-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.poisson-Tuple{Any, Any}"},[e("span",{class:"jlbinding"},"IncompressibleNavierStokes.poisson")],-1)),s[9]||(s[9]=i()),l(t,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[10]||(s[10]=a('
julia
poisson(psolver, f) -> Any
Solve the Poisson equation for the pressure with right hand side f at time t. For periodic and no-slip BC, the sum of f should be zero.
Differentiable version.
source
',4))]),e("details",Q,[e("summary",null,[s[11]||(s[11]=e("a",{id:"IncompressibleNavierStokes.pressure!-NTuple{5, Any}",href:"#IncompressibleNavierStokes.pressure!-NTuple{5, Any}"},[e("span",{class:"jlbinding"},"IncompressibleNavierStokes.pressure!")],-1)),s[12]||(s[12]=i()),l(t,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[13]||(s[13]=a('
julia
pressure!(p, u, temp, t, setup; psolver, F)
Compute pressure from velocity field (in-place version).
source
',3))]),e("details",m,[e("summary",null,[s[14]||(s[14]=e("a",{id:"IncompressibleNavierStokes.pressure-NTuple{4, Any}",href:"#IncompressibleNavierStokes.pressure-NTuple{4, Any}"},[e("span",{class:"jlbinding"},"IncompressibleNavierStokes.pressure")],-1)),s[15]||(s[15]=i()),l(t,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[16]||(s[16]=a('
julia
pressure(u, temp, t, setup; psolver)
Compute pressure from velocity field. This makes the pressure compatible with the velocity field, resulting in same order pressure as velocity.
Differentiable version.
source
',4))]),e("details",T,[e("summary",null,[s[17]||(s[17]=e("a",{id:"IncompressibleNavierStokes.project!-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.project!-Tuple{Any, Any}"},[e("span",{class:"jlbinding"},"IncompressibleNavierStokes.project!")],-1)),s[18]||(s[18]=i()),l(t,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[19]||(s[19]=a('
julia
project!(u, setup; psolver, p)
Project velocity field onto divergence-free space (in-place version).
source
',3))]),e("details",b,[e("summary",null,[s[20]||(s[20]=e("a",{id:"IncompressibleNavierStokes.project-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.project-Tuple{Any, Any}"},[e("span",{class:"jlbinding"},"IncompressibleNavierStokes.project")],-1)),s[21]||(s[21]=i()),l(t,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[22]||(s[22]=a('
julia
project(u, setup; psolver)
Project velocity field onto divergence-free space (differentiable version).
source
',3))]),e("details",y,[e("summary",null,[s[23]||(s[23]=e("a",{id:"IncompressibleNavierStokes.psolver_cg-Tuple{Any}",href:"#IncompressibleNavierStokes.psolver_cg-Tuple{Any}"},[e("span",{class:"jlbinding"},"IncompressibleNavierStokes.psolver_cg")],-1)),s[24]||(s[24]=i()),l(t,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[25]||(s[25]=a(`
julia
psolver_cg(
+) -> Union{IncompressibleNavierStokes.var"#psolve!#101"{Bool}, IncompressibleNavierStokes.var"#psolve!#124"}
Get default Poisson solver from setup.
source
`,3))]),e("details",u,[e("summary",null,[s[5]||(s[5]=e("a",{id:"IncompressibleNavierStokes.poisson!-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.poisson!-Tuple{Any, Any}"},[e("span",{class:"jlbinding"},"IncompressibleNavierStokes.poisson!")],-1)),s[6]||(s[6]=i()),l(t,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[7]||(s[7]=a('
julia
poisson!(psolver, f) -> Any
Solve the Poisson equation for the pressure (in-place version).
source
',3))]),e("details",g,[e("summary",null,[s[8]||(s[8]=e("a",{id:"IncompressibleNavierStokes.poisson-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.poisson-Tuple{Any, Any}"},[e("span",{class:"jlbinding"},"IncompressibleNavierStokes.poisson")],-1)),s[9]||(s[9]=i()),l(t,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[10]||(s[10]=a('
julia
poisson(psolver, f) -> Any
Solve the Poisson equation for the pressure with right hand side f at time t. For periodic and no-slip BC, the sum of f should be zero.
Differentiable version.
source
',4))]),e("details",Q,[e("summary",null,[s[11]||(s[11]=e("a",{id:"IncompressibleNavierStokes.pressure!-NTuple{5, Any}",href:"#IncompressibleNavierStokes.pressure!-NTuple{5, Any}"},[e("span",{class:"jlbinding"},"IncompressibleNavierStokes.pressure!")],-1)),s[12]||(s[12]=i()),l(t,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[13]||(s[13]=a('
julia
pressure!(p, u, temp, t, setup; psolver, F)
Compute pressure from velocity field (in-place version).
source
',3))]),e("details",m,[e("summary",null,[s[14]||(s[14]=e("a",{id:"IncompressibleNavierStokes.pressure-NTuple{4, Any}",href:"#IncompressibleNavierStokes.pressure-NTuple{4, Any}"},[e("span",{class:"jlbinding"},"IncompressibleNavierStokes.pressure")],-1)),s[15]||(s[15]=i()),l(t,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[16]||(s[16]=a('
julia
pressure(u, temp, t, setup; psolver)
Compute pressure from velocity field. This makes the pressure compatible with the velocity field, resulting in same order pressure as velocity.
Differentiable version.
source
',4))]),e("details",T,[e("summary",null,[s[17]||(s[17]=e("a",{id:"IncompressibleNavierStokes.project!-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.project!-Tuple{Any, Any}"},[e("span",{class:"jlbinding"},"IncompressibleNavierStokes.project!")],-1)),s[18]||(s[18]=i()),l(t,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[19]||(s[19]=a('
julia
project!(u, setup; psolver, p)
Project velocity field onto divergence-free space (in-place version).
source
',3))]),e("details",b,[e("summary",null,[s[20]||(s[20]=e("a",{id:"IncompressibleNavierStokes.project-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.project-Tuple{Any, Any}"},[e("span",{class:"jlbinding"},"IncompressibleNavierStokes.project")],-1)),s[21]||(s[21]=i()),l(t,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[22]||(s[22]=a('
julia
project(u, setup; psolver)
Project velocity field onto divergence-free space (differentiable version).
source
',3))]),e("details",y,[e("summary",null,[s[23]||(s[23]=e("a",{id:"IncompressibleNavierStokes.psolver_cg-Tuple{Any}",href:"#IncompressibleNavierStokes.psolver_cg-Tuple{Any}"},[e("span",{class:"jlbinding"},"IncompressibleNavierStokes.psolver_cg")],-1)),s[24]||(s[24]=i()),l(t,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[25]||(s[25]=a(`
julia
psolver_cg(
     setup;
     abstol,
     reltol,
     maxiter,
     preconditioner
-) -> IncompressibleNavierStokes.var"#psolve!#111"{_A, _B, _C, IncompressibleNavierStokes.var"#laplace_diag#109"{ndrange, workgroupsize}} where {_A, _B, _C, ndrange, workgroupsize}
Conjugate gradients iterative Poisson solver.
source
`,3))]),e("details",v,[e("summary",null,[s[26]||(s[26]=e("a",{id:"IncompressibleNavierStokes.psolver_cg_matrix-Tuple{Any}",href:"#IncompressibleNavierStokes.psolver_cg_matrix-Tuple{Any}"},[e("span",{class:"jlbinding"},"IncompressibleNavierStokes.psolver_cg_matrix")],-1)),s[27]||(s[27]=i()),l(t,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[28]||(s[28]=a(`
julia
psolver_cg_matrix(
+) -> IncompressibleNavierStokes.var"#psolve!#111"{_A, _B, _C, IncompressibleNavierStokes.var"#laplace_diag#109"{ndrange, workgroupsize}} where {_A, _B, _C, ndrange, workgroupsize}
Conjugate gradients iterative Poisson solver.
source
`,3))]),e("details",v,[e("summary",null,[s[26]||(s[26]=e("a",{id:"IncompressibleNavierStokes.psolver_cg_matrix-Tuple{Any}",href:"#IncompressibleNavierStokes.psolver_cg_matrix-Tuple{Any}"},[e("span",{class:"jlbinding"},"IncompressibleNavierStokes.psolver_cg_matrix")],-1)),s[27]||(s[27]=i()),l(t,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[28]||(s[28]=a(`
julia
psolver_cg_matrix(
     setup;
     kwargs...
-) -> IncompressibleNavierStokes.var"#psolve!#105"{Base.Pairs{Symbol, Union{}, Tuple{}, @NamedTuple{}}}
Conjugate gradients iterative Poisson solver. The kwargs are passed to the cg! function from IterativeSolvers.jl.
source
`,3))]),e("details",E,[e("summary",null,[s[29]||(s[29]=e("a",{id:"IncompressibleNavierStokes.psolver_direct-Tuple{Any}",href:"#IncompressibleNavierStokes.psolver_direct-Tuple{Any}"},[e("span",{class:"jlbinding"},"IncompressibleNavierStokes.psolver_direct")],-1)),s[30]||(s[30]=i()),l(t,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[31]||(s[31]=a(`
julia
psolver_direct(
+) -> IncompressibleNavierStokes.var"#psolve!#105"{Base.Pairs{Symbol, Union{}, Tuple{}, @NamedTuple{}}}
Conjugate gradients iterative Poisson solver. The kwargs are passed to the cg! function from IterativeSolvers.jl.
source
`,3))]),e("details",E,[e("summary",null,[s[29]||(s[29]=e("a",{id:"IncompressibleNavierStokes.psolver_direct-Tuple{Any}",href:"#IncompressibleNavierStokes.psolver_direct-Tuple{Any}"},[e("span",{class:"jlbinding"},"IncompressibleNavierStokes.psolver_direct")],-1)),s[30]||(s[30]=i()),l(t,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[31]||(s[31]=a(`
julia
psolver_direct(
     setup
-) -> IncompressibleNavierStokes.var"#psolve!#101"{Bool}
Create direct Poisson solver using an appropriate matrix decomposition.
source
`,3))]),e("details",f,[e("summary",null,[s[32]||(s[32]=e("a",{id:"IncompressibleNavierStokes.psolver_spectral-Tuple{Any}",href:"#IncompressibleNavierStokes.psolver_spectral-Tuple{Any}"},[e("span",{class:"jlbinding"},"IncompressibleNavierStokes.psolver_spectral")],-1)),s[33]||(s[33]=i()),l(t,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[34]||(s[34]=a(`
julia
psolver_spectral(
+) -> IncompressibleNavierStokes.var"#psolve!#101"{Bool}
Create direct Poisson solver using an appropriate matrix decomposition.
source
`,3))]),e("details",f,[e("summary",null,[s[32]||(s[32]=e("a",{id:"IncompressibleNavierStokes.psolver_spectral-Tuple{Any}",href:"#IncompressibleNavierStokes.psolver_spectral-Tuple{Any}"},[e("span",{class:"jlbinding"},"IncompressibleNavierStokes.psolver_spectral")],-1)),s[33]||(s[33]=i()),l(t,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[34]||(s[34]=a(`
julia
psolver_spectral(
     setup
-) -> IncompressibleNavierStokes.var"#psolve!#124"
Create spectral Poisson solver from setup.
source
`,3))])])}const H=r(d,[["render",j]]);export{I as __pageData,H as default};
+) -> IncompressibleNavierStokes.var"#psolve!#124"
Create spectral Poisson solver from setup.
source
`,3))])])}const H=r(d,[["render",j]]);export{I as __pageData,H as default};
diff --git a/dev/assets/manual_pressure.md.CRdAlBfk.lean.js b/dev/assets/manual_pressure.md.CGOE2Wqh.lean.js
similarity index 95%
rename from dev/assets/manual_pressure.md.CRdAlBfk.lean.js
rename to dev/assets/manual_pressure.md.CGOE2Wqh.lean.js
index 5e80ac24a..b47c4639e 100644
--- a/dev/assets/manual_pressure.md.CRdAlBfk.lean.js
+++ b/dev/assets/manual_pressure.md.CGOE2Wqh.lean.js
@@ -1,16 +1,16 @@
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julia
default_psolver(
     setup
-) -> Union{IncompressibleNavierStokes.var"#psolve!#101"{Bool}, IncompressibleNavierStokes.var"#psolve!#124"}
Get default Poisson solver from setup.
source
`,3))]),e("details",u,[e("summary",null,[s[5]||(s[5]=e("a",{id:"IncompressibleNavierStokes.poisson!-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.poisson!-Tuple{Any, Any}"},[e("span",{class:"jlbinding"},"IncompressibleNavierStokes.poisson!")],-1)),s[6]||(s[6]=i()),l(t,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[7]||(s[7]=a('
julia
poisson!(psolver, f) -> Any
Solve the Poisson equation for the pressure (in-place version).
source
',3))]),e("details",g,[e("summary",null,[s[8]||(s[8]=e("a",{id:"IncompressibleNavierStokes.poisson-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.poisson-Tuple{Any, Any}"},[e("span",{class:"jlbinding"},"IncompressibleNavierStokes.poisson")],-1)),s[9]||(s[9]=i()),l(t,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[10]||(s[10]=a('
julia
poisson(psolver, f) -> Any
Solve the Poisson equation for the pressure with right hand side f at time t. For periodic and no-slip BC, the sum of f should be zero.
Differentiable version.
source
',4))]),e("details",Q,[e("summary",null,[s[11]||(s[11]=e("a",{id:"IncompressibleNavierStokes.pressure!-NTuple{5, Any}",href:"#IncompressibleNavierStokes.pressure!-NTuple{5, Any}"},[e("span",{class:"jlbinding"},"IncompressibleNavierStokes.pressure!")],-1)),s[12]||(s[12]=i()),l(t,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[13]||(s[13]=a('
julia
pressure!(p, u, temp, t, setup; psolver, F)
Compute pressure from velocity field (in-place version).
source
',3))]),e("details",m,[e("summary",null,[s[14]||(s[14]=e("a",{id:"IncompressibleNavierStokes.pressure-NTuple{4, Any}",href:"#IncompressibleNavierStokes.pressure-NTuple{4, Any}"},[e("span",{class:"jlbinding"},"IncompressibleNavierStokes.pressure")],-1)),s[15]||(s[15]=i()),l(t,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[16]||(s[16]=a('
julia
pressure(u, temp, t, setup; psolver)
Compute pressure from velocity field. This makes the pressure compatible with the velocity field, resulting in same order pressure as velocity.
Differentiable version.
source
',4))]),e("details",T,[e("summary",null,[s[17]||(s[17]=e("a",{id:"IncompressibleNavierStokes.project!-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.project!-Tuple{Any, Any}"},[e("span",{class:"jlbinding"},"IncompressibleNavierStokes.project!")],-1)),s[18]||(s[18]=i()),l(t,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[19]||(s[19]=a('
julia
project!(u, setup; psolver, p)
Project velocity field onto divergence-free space (in-place version).
source
',3))]),e("details",b,[e("summary",null,[s[20]||(s[20]=e("a",{id:"IncompressibleNavierStokes.project-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.project-Tuple{Any, Any}"},[e("span",{class:"jlbinding"},"IncompressibleNavierStokes.project")],-1)),s[21]||(s[21]=i()),l(t,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[22]||(s[22]=a('
julia
project(u, setup; psolver)
Project velocity field onto divergence-free space (differentiable version).
source
',3))]),e("details",y,[e("summary",null,[s[23]||(s[23]=e("a",{id:"IncompressibleNavierStokes.psolver_cg-Tuple{Any}",href:"#IncompressibleNavierStokes.psolver_cg-Tuple{Any}"},[e("span",{class:"jlbinding"},"IncompressibleNavierStokes.psolver_cg")],-1)),s[24]||(s[24]=i()),l(t,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[25]||(s[25]=a(`
julia
psolver_cg(
+) -> Union{IncompressibleNavierStokes.var"#psolve!#101"{Bool}, IncompressibleNavierStokes.var"#psolve!#124"}
Get default Poisson solver from setup.
source
`,3))]),e("details",u,[e("summary",null,[s[5]||(s[5]=e("a",{id:"IncompressibleNavierStokes.poisson!-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.poisson!-Tuple{Any, Any}"},[e("span",{class:"jlbinding"},"IncompressibleNavierStokes.poisson!")],-1)),s[6]||(s[6]=i()),l(t,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[7]||(s[7]=a('
julia
poisson!(psolver, f) -> Any
Solve the Poisson equation for the pressure (in-place version).
source
',3))]),e("details",g,[e("summary",null,[s[8]||(s[8]=e("a",{id:"IncompressibleNavierStokes.poisson-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.poisson-Tuple{Any, Any}"},[e("span",{class:"jlbinding"},"IncompressibleNavierStokes.poisson")],-1)),s[9]||(s[9]=i()),l(t,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[10]||(s[10]=a('
julia
poisson(psolver, f) -> Any
Solve the Poisson equation for the pressure with right hand side f at time t. For periodic and no-slip BC, the sum of f should be zero.
Differentiable version.
source
',4))]),e("details",Q,[e("summary",null,[s[11]||(s[11]=e("a",{id:"IncompressibleNavierStokes.pressure!-NTuple{5, Any}",href:"#IncompressibleNavierStokes.pressure!-NTuple{5, Any}"},[e("span",{class:"jlbinding"},"IncompressibleNavierStokes.pressure!")],-1)),s[12]||(s[12]=i()),l(t,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[13]||(s[13]=a('
julia
pressure!(p, u, temp, t, setup; psolver, F)
Compute pressure from velocity field (in-place version).
source
',3))]),e("details",m,[e("summary",null,[s[14]||(s[14]=e("a",{id:"IncompressibleNavierStokes.pressure-NTuple{4, Any}",href:"#IncompressibleNavierStokes.pressure-NTuple{4, Any}"},[e("span",{class:"jlbinding"},"IncompressibleNavierStokes.pressure")],-1)),s[15]||(s[15]=i()),l(t,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[16]||(s[16]=a('
julia
pressure(u, temp, t, setup; psolver)
Compute pressure from velocity field. This makes the pressure compatible with the velocity field, resulting in same order pressure as velocity.
Differentiable version.
source
',4))]),e("details",T,[e("summary",null,[s[17]||(s[17]=e("a",{id:"IncompressibleNavierStokes.project!-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.project!-Tuple{Any, Any}"},[e("span",{class:"jlbinding"},"IncompressibleNavierStokes.project!")],-1)),s[18]||(s[18]=i()),l(t,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[19]||(s[19]=a('
julia
project!(u, setup; psolver, p)
Project velocity field onto divergence-free space (in-place version).
source
',3))]),e("details",b,[e("summary",null,[s[20]||(s[20]=e("a",{id:"IncompressibleNavierStokes.project-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.project-Tuple{Any, Any}"},[e("span",{class:"jlbinding"},"IncompressibleNavierStokes.project")],-1)),s[21]||(s[21]=i()),l(t,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[22]||(s[22]=a('
julia
project(u, setup; psolver)
Project velocity field onto divergence-free space (differentiable version).
source
',3))]),e("details",y,[e("summary",null,[s[23]||(s[23]=e("a",{id:"IncompressibleNavierStokes.psolver_cg-Tuple{Any}",href:"#IncompressibleNavierStokes.psolver_cg-Tuple{Any}"},[e("span",{class:"jlbinding"},"IncompressibleNavierStokes.psolver_cg")],-1)),s[24]||(s[24]=i()),l(t,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[25]||(s[25]=a(`
julia
psolver_cg(
     setup;
     abstol,
     reltol,
     maxiter,
     preconditioner
-) -> IncompressibleNavierStokes.var"#psolve!#111"{_A, _B, _C, IncompressibleNavierStokes.var"#laplace_diag#109"{ndrange, workgroupsize}} where {_A, _B, _C, ndrange, workgroupsize}
Conjugate gradients iterative Poisson solver.
source
`,3))]),e("details",v,[e("summary",null,[s[26]||(s[26]=e("a",{id:"IncompressibleNavierStokes.psolver_cg_matrix-Tuple{Any}",href:"#IncompressibleNavierStokes.psolver_cg_matrix-Tuple{Any}"},[e("span",{class:"jlbinding"},"IncompressibleNavierStokes.psolver_cg_matrix")],-1)),s[27]||(s[27]=i()),l(t,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[28]||(s[28]=a(`
julia
psolver_cg_matrix(
+) -> IncompressibleNavierStokes.var"#psolve!#111"{_A, _B, _C, IncompressibleNavierStokes.var"#laplace_diag#109"{ndrange, workgroupsize}} where {_A, _B, _C, ndrange, workgroupsize}
Conjugate gradients iterative Poisson solver.
source
`,3))]),e("details",v,[e("summary",null,[s[26]||(s[26]=e("a",{id:"IncompressibleNavierStokes.psolver_cg_matrix-Tuple{Any}",href:"#IncompressibleNavierStokes.psolver_cg_matrix-Tuple{Any}"},[e("span",{class:"jlbinding"},"IncompressibleNavierStokes.psolver_cg_matrix")],-1)),s[27]||(s[27]=i()),l(t,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[28]||(s[28]=a(`
julia
psolver_cg_matrix(
     setup;
     kwargs...
-) -> IncompressibleNavierStokes.var"#psolve!#105"{Base.Pairs{Symbol, Union{}, Tuple{}, @NamedTuple{}}}
Conjugate gradients iterative Poisson solver. The kwargs are passed to the cg! function from IterativeSolvers.jl.
source
`,3))]),e("details",E,[e("summary",null,[s[29]||(s[29]=e("a",{id:"IncompressibleNavierStokes.psolver_direct-Tuple{Any}",href:"#IncompressibleNavierStokes.psolver_direct-Tuple{Any}"},[e("span",{class:"jlbinding"},"IncompressibleNavierStokes.psolver_direct")],-1)),s[30]||(s[30]=i()),l(t,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[31]||(s[31]=a(`
julia
psolver_direct(
+) -> IncompressibleNavierStokes.var"#psolve!#105"{Base.Pairs{Symbol, Union{}, Tuple{}, @NamedTuple{}}}
Conjugate gradients iterative Poisson solver. The kwargs are passed to the cg! function from IterativeSolvers.jl.
source
`,3))]),e("details",E,[e("summary",null,[s[29]||(s[29]=e("a",{id:"IncompressibleNavierStokes.psolver_direct-Tuple{Any}",href:"#IncompressibleNavierStokes.psolver_direct-Tuple{Any}"},[e("span",{class:"jlbinding"},"IncompressibleNavierStokes.psolver_direct")],-1)),s[30]||(s[30]=i()),l(t,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[31]||(s[31]=a(`
julia
psolver_direct(
     setup
-) -> IncompressibleNavierStokes.var"#psolve!#101"{Bool}
Create direct Poisson solver using an appropriate matrix decomposition.
source
`,3))]),e("details",f,[e("summary",null,[s[32]||(s[32]=e("a",{id:"IncompressibleNavierStokes.psolver_spectral-Tuple{Any}",href:"#IncompressibleNavierStokes.psolver_spectral-Tuple{Any}"},[e("span",{class:"jlbinding"},"IncompressibleNavierStokes.psolver_spectral")],-1)),s[33]||(s[33]=i()),l(t,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[34]||(s[34]=a(`
julia
psolver_spectral(
+) -> IncompressibleNavierStokes.var"#psolve!#101"{Bool}
Create direct Poisson solver using an appropriate matrix decomposition.
source
`,3))]),e("details",f,[e("summary",null,[s[32]||(s[32]=e("a",{id:"IncompressibleNavierStokes.psolver_spectral-Tuple{Any}",href:"#IncompressibleNavierStokes.psolver_spectral-Tuple{Any}"},[e("span",{class:"jlbinding"},"IncompressibleNavierStokes.psolver_spectral")],-1)),s[33]||(s[33]=i()),l(t,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[34]||(s[34]=a(`
julia
psolver_spectral(
     setup
-) -> IncompressibleNavierStokes.var"#psolve!#124"
Create spectral Poisson solver from setup.
source
`,3))])])}const H=r(d,[["render",j]]);export{I as __pageData,H as default};
+) -> IncompressibleNavierStokes.var"#psolve!#124"
Create spectral Poisson solver from setup.
source
`,3))])])}const H=r(d,[["render",j]]);export{I as __pageData,H as default};
diff --git a/dev/assets/manual_sciml.md.CER13ld6.js b/dev/assets/manual_sciml.md.CcNBgquF.js
similarity index 75%
rename from dev/assets/manual_sciml.md.CER13ld6.js
rename to dev/assets/manual_sciml.md.CcNBgquF.js
index 31cefc7c6..4f8747521 100644
--- a/dev/assets/manual_sciml.md.CER13ld6.js
+++ b/dev/assets/manual_sciml.md.CcNBgquF.js
@@ -9,48 +9,48 @@ import{_ as T,c as i,j as a,a as t,a5 as n,G as Q,B as p,o as e}from"./chunks/fr
 problem = ODEProblem(f, u0, tspan) #SciMLBase.ODEProblem
 sol = solve(problem, Tsit5(), reltol = 1e-8, abstol = 1e-8) # sol: SciMLBase.ODESolution
retcode: Success
 Interpolation: specialized 4th order "free" interpolation
-t: 231-element Vector{Float64}:
+t: 204-element Vector{Float64}:
  0.0
- 0.0016659277128752055
- 0.002432989100957199
- 0.0036642837413415376
- 0.004709119986944817
- 0.005918078594056992
- 0.007106129763889296
- 0.00836604328705504
- 0.00964424423128464
- 0.010963315612198047
+ 0.0017630725463301349
+ 0.002578765391583634
+ 0.0038900689003174543
+ 0.0050004385667972275
+ 0.006282968599569631
+ 0.00753879858193828
+ 0.008865534083334846
+ 0.01020691087682567
+ 0.011586923053397053
  ⋮
- 0.901224250082742
- 0.9130080474847379
- 0.9249889455158832
- 0.9371708873148851
- 0.9495567173255886
- 0.9621483766861515
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+ 0.9327544152402584
+ 0.9463937370590901
+ 0.9602625548035638
+ 0.9743640183560942
+ 0.9887005249941152
  1.0
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+u: 204-element Vector{Array{Float64, 3}}:
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+ [0.37075720835548737 -0.12557069044081098 … 0.37075720835548737 -0.12557069044081098; -0.07440002612503235 -0.5385623982276309 … -0.1719243514619671 -0.41973786120245304; … ; 0.37075720835548737 -0.26217513977719487 … 0.24453144804547416 -0.12557069044081098; -0.07440002612503235 -0.41973786120245304 … -0.07440002612503235 -0.41973786120245304;;; 0.30173707668705024 0.6734710804676622 … 0.30173707668705024 0.6734710804676622; 0.5370511513252602 1.0717162458898755 … 0.795328987439447 0.9657206624363858; … ; 0.30173707668705024 0.8103888188649462 … 0.5703936018276079 0.6734710804676622; 0.5370511513252602 0.9657206624363858 … 0.5370511513252602 0.9657206624363858]
  ⋮
- [-2.6008269440350635 -2.6707555983699267 … -2.6008269440350635 -2.6707555983699267; -1.7530707144997295 0.09689754026634612 … 0.08931652432247436 -1.5878474701844734; … ; -2.6008269440350635 0.10720478554028837 … 0.10270212910901891 -2.6707555983699267; -1.7530707144997295 -1.5878474701844734 … -1.7530707144997295 -1.5878474701844734;;; -0.33445088457939787 0.7427051143709156 … -0.33445088457939787 0.7427051143709156; -0.3997760803661724 0.49976125856316267 … 0.48945401328925586 1.0678539731154677; … ; -0.33445088457939787 0.49375215219778495 … 0.48034789830989794 0.7427051143709156; -0.3997760803661724 1.0678539731154677 … -0.3997760803661724 1.0678539731154677]
- [-2.601725136184899 -2.670873685384534 … -2.601725136184899 -2.670873685384534; -1.7563563398627091 0.10432541979422275 … 0.09676710645672201 -1.5905130283970155; … ; -2.601725136184899 0.11653309235413584 … 0.11187293004895257 -2.670873685384534; -1.7563563398627091 -1.5905130283970155 … -1.7563563398627091 -1.5905130283970155;;; -0.3522441846097827 0.7271704846904127 … -0.3522441846097827 0.7271704846904127; -0.41329173959034654 0.49289307311066627 … 0.48068540055078907 1.056633316058722; … ; -0.3522441846097827 0.48307344888160403 … 0.46792424402653476 0.7271704846904127; -0.41329173959034654 1.056633316058722 … -0.41329173959034654 1.056633316058722]
- [-2.6025523523428413 -2.67071744709628 … -2.6025523523428413 -2.67071744709628; -1.7595072686400752 0.11170467403179123 … 0.10398180790626674 -1.592875884389524; … ; -2.6025523523428413 0.12573523533040898 … 0.12069919884062454 -2.67071744709628; -1.7595072686400752 -1.592875884389524 … -1.7595072686400752 -1.592875884389524;;; -0.3716771235605255 0.7097434315940944 … -0.3716771235605255 0.7097434315940944; -0.42842794202681916 0.4845376104979045 … 0.470507049199323 1.04360335395341; … ; -0.3716771235605255 0.4708320861058763 … 0.4540567364223937 0.7097434315940944; -0.42842794202681916 1.04360335395341 … -0.42842794202681916 1.04360335395341]
- [-2.6033919710125044 -2.6703849125869374 … -2.6033919710125044 -2.6703849125869374; -1.7625884932762157 0.11894958866689538 … 0.1108737142172876 -1.595010124997473; … ; -2.6033919710125044 0.1346989359254884 … 0.12906953547026517 -2.6703849125869374; -1.7625884932762157 -1.595010124997473 … -1.7625884932762157 -1.595010124997473;;; -0.3926613103435367 0.6904669028807235 … -0.3926613103435367 0.6904669028807235; -0.44514969973593976 0.4746627267546408 … 0.45891337949608374 1.0287556320125302; … ; -0.3926613103435367 0.4570355653834647 … 0.43877942685883076 0.6904669028807235; -0.44514969973593976 1.0287556320125302 … -0.44514969973593976 1.0287556320125302]
- [-2.6043280594687412 -2.6699787184933967 … -2.6043280594687412 -2.6699787184933967; -1.7656684357322596 0.12596507497113008 … 0.11734810375345404 -1.5969962068782388; … ; -2.6043280594687412 0.14330182663680066 … 0.13686518330313233 -2.6699787184933967; -1.7656684357322596 -1.5969962068782388 … -1.7656684357322596 -1.5969962068782388;;; -0.4150660872718007 0.6694280502383273 … -0.4150660872718007 0.6694280502383273; -0.4633875966731134 0.4632574290511748 … 0.44592067738554064 1.0121162640308192; … ; -0.4150660872718007 0.44171996769365324 … 0.4221548844285295 0.6694280502383273; -0.4633875966731134 1.0121162640308192 … -0.4633875966731134 1.0121162640308192]
- [-2.60544291809869 -2.669603929032186 … -2.60544291809869 -2.669603929032186; -1.7688174940989976 0.1326468336290909 … 0.12330344096559884 -1.5989199688668567; … ; -2.60544291809869 0.15141245391774916 … 0.14396194354418868 -2.669603929032186; -1.7688174940989976 -1.5989199688668567 … -1.7688174940989976 -1.5989199688668567;;; -0.4387165340878582 0.6467626475342549 … -0.4387165340878582 0.6467626475342549; -0.48303489816549916 0.4503361141495976 … 0.4315704938609758 0.9937508037134614; … ; -0.4387165340878582 0.42495415887972643 … 0.4042773366315602 0.6467626475342549; -0.48303489816549916 0.9937508037134614 … -0.48303489816549916 0.9937508037134614]
- [-2.6068139332770977 -2.6693650234422095 … -2.6068139332770977 -2.6693650234422095; -1.77210571029155 0.13888179342208257 … 0.12863269582615977 -1.6008706855176826; … ; -2.6068139332770977 0.15889167346612212 … 0.15023250924374582 -2.6693650234422095; -1.77210571029155 -1.6008706855176826 … -1.77210571029155 -1.6008706855176826;;; -0.463391491674076 0.6226597041634463 … -0.463391491674076 0.6226597041634463; -0.5039442703140851 0.4359437228387498 … 0.4159338427947474 0.973769704494905; … ; -0.463391491674076 0.40684472968608854 … 0.385276367047268 0.6226597041634463; -0.5039442703140851 0.973769704494905 … -0.5039442703140851 0.973769704494905]
- [-2.608510095624186 -2.6693620397707933 … -2.608510095624186 -2.6693620397707933; -1.7756002089136793 0.14455026801749912 … 0.13322622135795462 -1.6029384976802836; … ; -2.608510095624186 0.1655961649508638 … 0.15555064935246116 -2.6693620397707933; -1.7756002089136793 -1.6029384976802836 … -1.7756002089136793 -1.6029384976802836;;; -0.4888266014764763 0.5973613484526877 … -0.4888266014764763 0.5973613484526877; -0.5259282049009588 0.4201590976985429 … 0.3991132007652158 0.952330459812817; … ; -0.4888266014764763 0.3875380448857541 … 0.365317344444688 0.5973613484526877; -0.5259282049009588 0.952330459812817 … -0.5259282049009588 0.952330459812817]
- [-2.610391429817098 -2.669642952553338 … -2.610391429817098 -2.669642952553338; -1.7790232145791847 0.14912668082348163 … 0.1366855419132724 -1.605003300278285; … ; -2.610391429817098 0.17091984605417793 … 0.15947240646934602 -2.669642952553338; -1.7790232145791847 -1.605003300278285 … -1.7790232145791847 -1.605003300278285;;; -0.512459536652462 0.5734647082450985 … -0.512459536652462 0.5734647082450985; -0.5467494943080825 0.4046273175965032 … 0.3828341523658447 0.9316537922498789; … ; -0.512459536652462 0.3690211337021703 … 0.3464251477003929 0.5734647082450985; -0.5467494943080825 0.9316537922498789 … -0.5467494943080825 0.9316537922498789]
Alternatively, it is also possible to use an in-place formulation
julia
f!(du,u,p,t) = right_hand_side!(du, u, Ref([setup, psolver]), t)
+ [3.63658584761423 3.87080429663847 … 3.63658584761423 3.87080429663847; 2.6792423852120266 -0.1225160105596109 … -0.10545067559959928 2.802014516816849; … ; 3.63658584761423 -0.12912223894278943 … -0.11638439519314522 3.87080429663847; 2.6792423852120266 2.802014516816849 … 2.6792423852120266 2.802014516816849;;; -3.7994309843333527 -4.208021775307386 … -3.7994309843333527 -4.208021775307386; -3.5271355479570685 0.0368409287211739 … 0.04344715710435443 -4.001965753158543; … ; -3.7994309843333527 0.014947618503139795 … 0.018108013501957024 -4.208021775307386; -3.5271355479570685 -4.001965753158543 … -3.5271355479570685 -4.001965753158543]
+ [3.6422895086948026 3.8754147141679445 … 3.6422895086948026 3.8754147141679445; 2.6861592819208227 -0.1102895726892691 … -0.09162669561786173 2.807686775420112; … ; 3.6422895086948026 -0.1177386254998089 … -0.10346473297439138 3.8754147141679445; 2.6861592819208227 2.807686775420112 … 2.6861592819208227 2.807686775420112;;; -3.7931619925313242 -4.202317958546451 … -3.7931619925313242 -4.202317958546451; -3.520066837125464 0.03622302471012847 … 0.04367207752067032 -3.995334854799291; … ; -3.7931619925313242 0.013948554503623277 … 0.01797602011337264 -4.202317958546451; -3.520066837125464 -3.995334854799291 … -3.520066837125464 -3.995334854799291]
+ [3.6479937698880462 3.880149347154071 … 3.6479937698880462 3.880149347154071; 2.6930364840785233 -0.09783809929702272 … -0.07776985332212666 2.813469779247899; … ; 3.6479937698880462 -0.10613939385057908 … -0.09048860694780399 3.880149347154071; 2.6930364840785233 2.813469779247899 … 2.6930364840785233 2.813469779247899;;; -3.787054128421903 -4.19668244159331 … -3.787054128421903 -4.19668244159331; -3.5132283960223627 0.03581805063493754 … 0.04411934518849587 -3.988821699344505; … ; -3.787054128421903 0.013198507566411128 … 0.01809997695775903 -4.19668244159331; -3.5132283960223627 -3.988821699344505 … -3.5132283960223627 -3.988821699344505]
+ [3.653648807156927 3.884965007263431 … 3.653648807156927 3.884965007263431; 2.699812752232953 -0.0852337491691666 … -0.0639660324794772 2.819309423233572; … ; 3.653648807156927 -0.09438466632096755 … -0.07752884716293337 3.884965007263431; 2.699812752232953 2.819309423233572 … 2.699812752232953 2.819309423233572;;; -3.781138941703361 -4.191145491275563 … -3.781138941703361 -4.191145491275563; -3.506658733142349 0.03562880939265571 … 0.044779726544458685 -3.9824637292406186; … ; -3.781138941703361 0.012699036170763192 … 0.01847153724798791 -4.191145491275563; -3.506658733142349 -3.9824637292406186 … -3.506658733142349 -3.9824637292406186]
+ [3.659205979536996 3.889816751078532 … 3.659205979536996 3.889816751078532; 2.7064289741897443 -0.07255414846164322 … -0.05030101815559454 2.825149688747564; … ; 3.659205979536996 -0.08253980267461902 … -0.06465938047302927 3.889816751078532; 2.7064289741897443 2.825149688747564 … 2.7064289741897443 2.825149688747564;;; -3.775446647182436 -4.185737271493904 … -3.775446647182436 -4.185737271493904; -3.5003941254461775 0.03565465237106255 … 0.04564030658404039 -3.976298023468234; … ; -3.775446647182436 0.012448758810299665 … 0.019078978590910903 -4.185737271493904; -3.5003941254461775 -3.976298023468234 … -3.5003941254461775 -3.976298023468234]
+ [3.664618686922103 3.8946587976801346 … 3.664618686922103 3.8946587976801346; 2.7128292402547007 -0.05988085958027444 … -0.0368592383448486 2.8309339928553903; … ; 3.664618686922103 -0.07067439363603704 … -0.05195426801286364 3.8946587976801346; 2.7128292402547007 2.8309339928553903 … 2.7128292402547007 2.8309339928553903;;; -3.7700058000739514 -4.180487610093032 … -3.7700058000739514 -4.180487610093032; -3.4944680861051314 0.0358913760423405 … 0.046684910098105145 -3.97036081328936; … ; -3.7700058000739514 0.012443154773298683 … 0.019907432510493542 -4.180487610093032; -3.4944680861051314 -3.97036081328936 … -3.4944680861051314 -3.97036081328936]
+ [3.669843653298168 3.8994459366486214 … 3.669843653298168 3.8994459366486214; 2.7189623589035268 -0.04729660729185216 … -0.023721384830430667 2.836607060649013; … ; 3.669843653298168 -0.05886002218388251 … -0.03948559259932328 3.8994459366486214; 2.7189623589035268 2.836607060649013 … 2.7189623589035268 2.836607060649013;;; -3.7648424333013457 -4.175425182695787 … -3.7648424333013457 -4.175425182695787; -3.4889102867335025 0.03633131410140703 … 0.04789472899343934 -3.964686349775314; … ; -3.7648424333013457 0.012674494032257701 … 0.020939295360611808 -4.175425182695787; -3.4889102867335025 -3.964686349775314 … -3.4889102867335025 -3.964686349775314]
+ [3.674841850453165 3.904134731547883 … 3.674841850453165 3.904134731547883; 2.724782882886711 -0.03488300018573105 … -0.010962759787617392 2.8421164708795263; … ; 3.674841850453165 -0.04716839537064069 … -0.027321933899741135 3.904134731547883; 2.724782882886711 2.8421164708795263 … 2.724782882886711 2.8421164708795263;;; -3.759979480483631 -4.170576881038018 … -3.759979480483631 -4.170576881038018; -3.4837458594845088 0.036963640271927826 … 0.049249035456839486 -3.959306024454923; … ; -3.759979480483631 0.01313197193049889 … 0.022154757023338023 -4.170576881038018; -3.4837458594845088 -3.959306024454923 … -3.4837458594845088 -3.959306024454923]
+ [3.6785458556973847 3.9076829270352 … 3.6785458556973847 3.9076829270352; 2.7290632367754313 -0.02540851001379519 … -0.0013580255384598883 2.8462520790618577; … ; 3.6785458556973847 -0.0382162612055059 … -0.01812500342619521 3.9076829270352; 2.7290632367754313 2.8462520790618577 … 2.7290632367754313 2.8462520790618577;;; -3.7564219120903566 -4.1669743965658705 … -3.7564219120903566 -4.1669743965658705; -3.4800187689780997 0.037579010665183425 … 0.05038676185689626 -3.9553476091080677; … ; -3.7564219120903566 0.013634784336917961 … 0.023211947861768473 -4.1669743965658705; -3.4800187689780997 -3.9553476091080677 … -3.4800187689780997 -3.9553476091080677]
Alternatively, it is also possible to use an in-place formulation
julia
f!(du,u,p,t) = right_hand_side!(du, u, Ref([setup, psolver]), t)
 u = similar(u0)
 du = similar(u0)
 p = nothing
@@ -58,4 +58,4 @@ import{_ as T,c as i,j as a,a as t,a5 as n,G as Q,B as p,o as e}from"./chunks/fr
 f!(du,u,p,t)
that is usually faster than the out-of-place formulation.
You can look here for more information on how to use the SciML solvers and all the options available.
API ​
`,10)),a("details",g,[a("summary",null,[s[8]||(s[8]=a("a",{id:"IncompressibleNavierStokes.create_right_hand_side-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.create_right_hand_side-Tuple{Any, Any}"},[a("span",{class:"jlbinding"},"IncompressibleNavierStokes.create_right_hand_side")],-1)),s[9]||(s[9]=t()),Q(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[10]||(s[10]=n(`
julia
create_right_hand_side(
     setup,
     psolver
-) -> IncompressibleNavierStokes.var"#right_hand_side#343"
create_right_hand_side(setup, psolver)
Creates a function that computes the right-hand side of the Navier-Stokes equations for a given setup and pressure solver.
Arguments
  • setup: The simulation setup containing grid and boundary conditions.
  • psolver: The pressure solver to be used.
Returns
A function that computes the right-hand side of the Navier-Stokes equations.
source
`,8))]),a("details",k,[a("summary",null,[s[11]||(s[11]=a("a",{id:"IncompressibleNavierStokes.right_hand_side!-NTuple{4, Any}",href:"#IncompressibleNavierStokes.right_hand_side!-NTuple{4, Any}"},[a("span",{class:"jlbinding"},"IncompressibleNavierStokes.right_hand_side!")],-1)),s[12]||(s[12]=t()),Q(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[13]||(s[13]=n('
julia
right_hand_side!(dudt, u, params_ref, t)
right_hand_side!(dudt, u, params_ref, t)
Computes the right-hand side of the Navier-Stokes equations in-place.
Arguments
  • dudt: The array to store the computed right-hand side.
  • u: The current velocity field.
  • params_ref: A reference to the parameters containing the setup and pressure solver.
  • t: The current time.
Returns
Nothing. The result is stored in dudt.
source
',8))])])}const v=T(r,[["render",c]]);export{b as __pageData,v as default};
+) -> IncompressibleNavierStokes.var"#right_hand_side#343"
create_right_hand_side(setup, psolver)
Creates a function that computes the right-hand side of the Navier-Stokes equations for a given setup and pressure solver.
Arguments
  • setup: The simulation setup containing grid and boundary conditions.
  • psolver: The pressure solver to be used.
Returns
A function that computes the right-hand side of the Navier-Stokes equations.
source
`,8))]),a("details",k,[a("summary",null,[s[11]||(s[11]=a("a",{id:"IncompressibleNavierStokes.right_hand_side!-NTuple{4, Any}",href:"#IncompressibleNavierStokes.right_hand_side!-NTuple{4, Any}"},[a("span",{class:"jlbinding"},"IncompressibleNavierStokes.right_hand_side!")],-1)),s[12]||(s[12]=t()),Q(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[13]||(s[13]=n('
julia
right_hand_side!(dudt, u, params_ref, t)
right_hand_side!(dudt, u, params_ref, t)
Computes the right-hand side of the Navier-Stokes equations in-place.
Arguments
  • dudt: The array to store the computed right-hand side.
  • u: The current velocity field.
  • params_ref: A reference to the parameters containing the setup and pressure solver.
  • t: The current time.
Returns
Nothing. The result is stored in dudt.
source
',8))])])}const v=T(r,[["render",c]]);export{b as __pageData,v as default};
diff --git a/dev/assets/manual_sciml.md.CER13ld6.lean.js b/dev/assets/manual_sciml.md.CcNBgquF.lean.js
similarity index 75%
rename from dev/assets/manual_sciml.md.CER13ld6.lean.js
rename to dev/assets/manual_sciml.md.CcNBgquF.lean.js
index 31cefc7c6..4f8747521 100644
--- a/dev/assets/manual_sciml.md.CER13ld6.lean.js
+++ b/dev/assets/manual_sciml.md.CcNBgquF.lean.js
@@ -9,48 +9,48 @@ import{_ as T,c as i,j as a,a as t,a5 as n,G as Q,B as p,o as e}from"./chunks/fr
 problem = ODEProblem(f, u0, tspan) #SciMLBase.ODEProblem
 sol = solve(problem, Tsit5(), reltol = 1e-8, abstol = 1e-8) # sol: SciMLBase.ODESolution
retcode: Success
 Interpolation: specialized 4th order "free" interpolation
-t: 231-element Vector{Float64}:
+t: 204-element Vector{Float64}:
  0.0
- 0.0016659277128752055
- 0.002432989100957199
- 0.0036642837413415376
- 0.004709119986944817
- 0.005918078594056992
- 0.007106129763889296
- 0.00836604328705504
- 0.00964424423128464
- 0.010963315612198047
+ 0.0017630725463301349
+ 0.002578765391583634
+ 0.0038900689003174543
+ 0.0050004385667972275
+ 0.006282968599569631
+ 0.00753879858193828
+ 0.008865534083334846
+ 0.01020691087682567
+ 0.011586923053397053
  ⋮
- 0.901224250082742
- 0.9130080474847379
- 0.9249889455158832
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- 0.9495567173255886
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- [-2.6043280594687412 -2.6699787184933967 … -2.6043280594687412 -2.6699787184933967; -1.7656684357322596 0.12596507497113008 … 0.11734810375345404 -1.5969962068782388; … ; -2.6043280594687412 0.14330182663680066 … 0.13686518330313233 -2.6699787184933967; -1.7656684357322596 -1.5969962068782388 … -1.7656684357322596 -1.5969962068782388;;; -0.4150660872718007 0.6694280502383273 … -0.4150660872718007 0.6694280502383273; -0.4633875966731134 0.4632574290511748 … 0.44592067738554064 1.0121162640308192; … ; -0.4150660872718007 0.44171996769365324 … 0.4221548844285295 0.6694280502383273; -0.4633875966731134 1.0121162640308192 … -0.4633875966731134 1.0121162640308192]
- [-2.60544291809869 -2.669603929032186 … -2.60544291809869 -2.669603929032186; -1.7688174940989976 0.1326468336290909 … 0.12330344096559884 -1.5989199688668567; … ; -2.60544291809869 0.15141245391774916 … 0.14396194354418868 -2.669603929032186; -1.7688174940989976 -1.5989199688668567 … -1.7688174940989976 -1.5989199688668567;;; -0.4387165340878582 0.6467626475342549 … -0.4387165340878582 0.6467626475342549; -0.48303489816549916 0.4503361141495976 … 0.4315704938609758 0.9937508037134614; … ; -0.4387165340878582 0.42495415887972643 … 0.4042773366315602 0.6467626475342549; -0.48303489816549916 0.9937508037134614 … -0.48303489816549916 0.9937508037134614]
- [-2.6068139332770977 -2.6693650234422095 … -2.6068139332770977 -2.6693650234422095; -1.77210571029155 0.13888179342208257 … 0.12863269582615977 -1.6008706855176826; … ; -2.6068139332770977 0.15889167346612212 … 0.15023250924374582 -2.6693650234422095; -1.77210571029155 -1.6008706855176826 … -1.77210571029155 -1.6008706855176826;;; -0.463391491674076 0.6226597041634463 … -0.463391491674076 0.6226597041634463; -0.5039442703140851 0.4359437228387498 … 0.4159338427947474 0.973769704494905; … ; -0.463391491674076 0.40684472968608854 … 0.385276367047268 0.6226597041634463; -0.5039442703140851 0.973769704494905 … -0.5039442703140851 0.973769704494905]
- [-2.608510095624186 -2.6693620397707933 … -2.608510095624186 -2.6693620397707933; -1.7756002089136793 0.14455026801749912 … 0.13322622135795462 -1.6029384976802836; … ; -2.608510095624186 0.1655961649508638 … 0.15555064935246116 -2.6693620397707933; -1.7756002089136793 -1.6029384976802836 … -1.7756002089136793 -1.6029384976802836;;; -0.4888266014764763 0.5973613484526877 … -0.4888266014764763 0.5973613484526877; -0.5259282049009588 0.4201590976985429 … 0.3991132007652158 0.952330459812817; … ; -0.4888266014764763 0.3875380448857541 … 0.365317344444688 0.5973613484526877; -0.5259282049009588 0.952330459812817 … -0.5259282049009588 0.952330459812817]
- [-2.610391429817098 -2.669642952553338 … -2.610391429817098 -2.669642952553338; -1.7790232145791847 0.14912668082348163 … 0.1366855419132724 -1.605003300278285; … ; -2.610391429817098 0.17091984605417793 … 0.15947240646934602 -2.669642952553338; -1.7790232145791847 -1.605003300278285 … -1.7790232145791847 -1.605003300278285;;; -0.512459536652462 0.5734647082450985 … -0.512459536652462 0.5734647082450985; -0.5467494943080825 0.4046273175965032 … 0.3828341523658447 0.9316537922498789; … ; -0.512459536652462 0.3690211337021703 … 0.3464251477003929 0.5734647082450985; -0.5467494943080825 0.9316537922498789 … -0.5467494943080825 0.9316537922498789]
Alternatively, it is also possible to use an in-place formulation
julia
f!(du,u,p,t) = right_hand_side!(du, u, Ref([setup, psolver]), t)
+ [3.63658584761423 3.87080429663847 … 3.63658584761423 3.87080429663847; 2.6792423852120266 -0.1225160105596109 … -0.10545067559959928 2.802014516816849; … ; 3.63658584761423 -0.12912223894278943 … -0.11638439519314522 3.87080429663847; 2.6792423852120266 2.802014516816849 … 2.6792423852120266 2.802014516816849;;; -3.7994309843333527 -4.208021775307386 … -3.7994309843333527 -4.208021775307386; -3.5271355479570685 0.0368409287211739 … 0.04344715710435443 -4.001965753158543; … ; -3.7994309843333527 0.014947618503139795 … 0.018108013501957024 -4.208021775307386; -3.5271355479570685 -4.001965753158543 … -3.5271355479570685 -4.001965753158543]
+ [3.6422895086948026 3.8754147141679445 … 3.6422895086948026 3.8754147141679445; 2.6861592819208227 -0.1102895726892691 … -0.09162669561786173 2.807686775420112; … ; 3.6422895086948026 -0.1177386254998089 … -0.10346473297439138 3.8754147141679445; 2.6861592819208227 2.807686775420112 … 2.6861592819208227 2.807686775420112;;; -3.7931619925313242 -4.202317958546451 … -3.7931619925313242 -4.202317958546451; -3.520066837125464 0.03622302471012847 … 0.04367207752067032 -3.995334854799291; … ; -3.7931619925313242 0.013948554503623277 … 0.01797602011337264 -4.202317958546451; -3.520066837125464 -3.995334854799291 … -3.520066837125464 -3.995334854799291]
+ [3.6479937698880462 3.880149347154071 … 3.6479937698880462 3.880149347154071; 2.6930364840785233 -0.09783809929702272 … -0.07776985332212666 2.813469779247899; … ; 3.6479937698880462 -0.10613939385057908 … -0.09048860694780399 3.880149347154071; 2.6930364840785233 2.813469779247899 … 2.6930364840785233 2.813469779247899;;; -3.787054128421903 -4.19668244159331 … -3.787054128421903 -4.19668244159331; -3.5132283960223627 0.03581805063493754 … 0.04411934518849587 -3.988821699344505; … ; -3.787054128421903 0.013198507566411128 … 0.01809997695775903 -4.19668244159331; -3.5132283960223627 -3.988821699344505 … -3.5132283960223627 -3.988821699344505]
+ [3.653648807156927 3.884965007263431 … 3.653648807156927 3.884965007263431; 2.699812752232953 -0.0852337491691666 … -0.0639660324794772 2.819309423233572; … ; 3.653648807156927 -0.09438466632096755 … -0.07752884716293337 3.884965007263431; 2.699812752232953 2.819309423233572 … 2.699812752232953 2.819309423233572;;; -3.781138941703361 -4.191145491275563 … -3.781138941703361 -4.191145491275563; -3.506658733142349 0.03562880939265571 … 0.044779726544458685 -3.9824637292406186; … ; -3.781138941703361 0.012699036170763192 … 0.01847153724798791 -4.191145491275563; -3.506658733142349 -3.9824637292406186 … -3.506658733142349 -3.9824637292406186]
+ [3.659205979536996 3.889816751078532 … 3.659205979536996 3.889816751078532; 2.7064289741897443 -0.07255414846164322 … -0.05030101815559454 2.825149688747564; … ; 3.659205979536996 -0.08253980267461902 … -0.06465938047302927 3.889816751078532; 2.7064289741897443 2.825149688747564 … 2.7064289741897443 2.825149688747564;;; -3.775446647182436 -4.185737271493904 … -3.775446647182436 -4.185737271493904; -3.5003941254461775 0.03565465237106255 … 0.04564030658404039 -3.976298023468234; … ; -3.775446647182436 0.012448758810299665 … 0.019078978590910903 -4.185737271493904; -3.5003941254461775 -3.976298023468234 … -3.5003941254461775 -3.976298023468234]
+ [3.664618686922103 3.8946587976801346 … 3.664618686922103 3.8946587976801346; 2.7128292402547007 -0.05988085958027444 … -0.0368592383448486 2.8309339928553903; … ; 3.664618686922103 -0.07067439363603704 … -0.05195426801286364 3.8946587976801346; 2.7128292402547007 2.8309339928553903 … 2.7128292402547007 2.8309339928553903;;; -3.7700058000739514 -4.180487610093032 … -3.7700058000739514 -4.180487610093032; -3.4944680861051314 0.0358913760423405 … 0.046684910098105145 -3.97036081328936; … ; -3.7700058000739514 0.012443154773298683 … 0.019907432510493542 -4.180487610093032; -3.4944680861051314 -3.97036081328936 … -3.4944680861051314 -3.97036081328936]
+ [3.669843653298168 3.8994459366486214 … 3.669843653298168 3.8994459366486214; 2.7189623589035268 -0.04729660729185216 … -0.023721384830430667 2.836607060649013; … ; 3.669843653298168 -0.05886002218388251 … -0.03948559259932328 3.8994459366486214; 2.7189623589035268 2.836607060649013 … 2.7189623589035268 2.836607060649013;;; -3.7648424333013457 -4.175425182695787 … -3.7648424333013457 -4.175425182695787; -3.4889102867335025 0.03633131410140703 … 0.04789472899343934 -3.964686349775314; … ; -3.7648424333013457 0.012674494032257701 … 0.020939295360611808 -4.175425182695787; -3.4889102867335025 -3.964686349775314 … -3.4889102867335025 -3.964686349775314]
+ [3.674841850453165 3.904134731547883 … 3.674841850453165 3.904134731547883; 2.724782882886711 -0.03488300018573105 … -0.010962759787617392 2.8421164708795263; … ; 3.674841850453165 -0.04716839537064069 … -0.027321933899741135 3.904134731547883; 2.724782882886711 2.8421164708795263 … 2.724782882886711 2.8421164708795263;;; -3.759979480483631 -4.170576881038018 … -3.759979480483631 -4.170576881038018; -3.4837458594845088 0.036963640271927826 … 0.049249035456839486 -3.959306024454923; … ; -3.759979480483631 0.01313197193049889 … 0.022154757023338023 -4.170576881038018; -3.4837458594845088 -3.959306024454923 … -3.4837458594845088 -3.959306024454923]
+ [3.6785458556973847 3.9076829270352 … 3.6785458556973847 3.9076829270352; 2.7290632367754313 -0.02540851001379519 … -0.0013580255384598883 2.8462520790618577; … ; 3.6785458556973847 -0.0382162612055059 … -0.01812500342619521 3.9076829270352; 2.7290632367754313 2.8462520790618577 … 2.7290632367754313 2.8462520790618577;;; -3.7564219120903566 -4.1669743965658705 … -3.7564219120903566 -4.1669743965658705; -3.4800187689780997 0.037579010665183425 … 0.05038676185689626 -3.9553476091080677; … ; -3.7564219120903566 0.013634784336917961 … 0.023211947861768473 -4.1669743965658705; -3.4800187689780997 -3.9553476091080677 … -3.4800187689780997 -3.9553476091080677]
Alternatively, it is also possible to use an in-place formulation
julia
f!(du,u,p,t) = right_hand_side!(du, u, Ref([setup, psolver]), t)
 u = similar(u0)
 du = similar(u0)
 p = nothing
@@ -58,4 +58,4 @@ import{_ as T,c as i,j as a,a as t,a5 as n,G as Q,B as p,o as e}from"./chunks/fr
 f!(du,u,p,t)
that is usually faster than the out-of-place formulation.
You can look here for more information on how to use the SciML solvers and all the options available.
API ​
`,10)),a("details",g,[a("summary",null,[s[8]||(s[8]=a("a",{id:"IncompressibleNavierStokes.create_right_hand_side-Tuple{Any, Any}",href:"#IncompressibleNavierStokes.create_right_hand_side-Tuple{Any, Any}"},[a("span",{class:"jlbinding"},"IncompressibleNavierStokes.create_right_hand_side")],-1)),s[9]||(s[9]=t()),Q(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[10]||(s[10]=n(`
julia
create_right_hand_side(
     setup,
     psolver
-) -> IncompressibleNavierStokes.var"#right_hand_side#343"
create_right_hand_side(setup, psolver)
Creates a function that computes the right-hand side of the Navier-Stokes equations for a given setup and pressure solver.
Arguments
  • setup: The simulation setup containing grid and boundary conditions.
  • psolver: The pressure solver to be used.
Returns
A function that computes the right-hand side of the Navier-Stokes equations.
source
`,8))]),a("details",k,[a("summary",null,[s[11]||(s[11]=a("a",{id:"IncompressibleNavierStokes.right_hand_side!-NTuple{4, Any}",href:"#IncompressibleNavierStokes.right_hand_side!-NTuple{4, Any}"},[a("span",{class:"jlbinding"},"IncompressibleNavierStokes.right_hand_side!")],-1)),s[12]||(s[12]=t()),Q(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[13]||(s[13]=n('
julia
right_hand_side!(dudt, u, params_ref, t)
right_hand_side!(dudt, u, params_ref, t)
Computes the right-hand side of the Navier-Stokes equations in-place.
Arguments
  • dudt: The array to store the computed right-hand side.
  • u: The current velocity field.
  • params_ref: A reference to the parameters containing the setup and pressure solver.
  • t: The current time.
Returns
Nothing. The result is stored in dudt.
source
',8))])])}const v=T(r,[["render",c]]);export{b as __pageData,v as default};
+) -> IncompressibleNavierStokes.var"#right_hand_side#343"
create_right_hand_side(setup, psolver)
Creates a function that computes the right-hand side of the Navier-Stokes equations for a given setup and pressure solver.
Arguments
  • setup: The simulation setup containing grid and boundary conditions.
  • psolver: The pressure solver to be used.
Returns
A function that computes the right-hand side of the Navier-Stokes equations.
source
`,8))]),a("details",k,[a("summary",null,[s[11]||(s[11]=a("a",{id:"IncompressibleNavierStokes.right_hand_side!-NTuple{4, Any}",href:"#IncompressibleNavierStokes.right_hand_side!-NTuple{4, Any}"},[a("span",{class:"jlbinding"},"IncompressibleNavierStokes.right_hand_side!")],-1)),s[12]||(s[12]=t()),Q(l,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[13]||(s[13]=n('
julia
right_hand_side!(dudt, u, params_ref, t)
right_hand_side!(dudt, u, params_ref, t)
Computes the right-hand side of the Navier-Stokes equations in-place.
Arguments
  • dudt: The array to store the computed right-hand side.
  • u: The current velocity field.
  • params_ref: A reference to the parameters containing the setup and pressure solver.
  • t: The current time.
Returns
Nothing. The result is stored in dudt.
source
',8))])])}const v=T(r,[["render",c]]);export{b as __pageData,v as default};
diff --git a/dev/assets/manual_setup.md.w6x6ynsm.js b/dev/assets/manual_setup.md.B7b6UkIH.js
similarity index 97%
rename from dev/assets/manual_setup.md.w6x6ynsm.js
rename to dev/assets/manual_setup.md.B7b6UkIH.js
index b1bac10f5..6084c86b4 100644
--- a/dev/assets/manual_setup.md.w6x6ynsm.js
+++ b/dev/assets/manual_setup.md.B7b6UkIH.js
@@ -1,10 +1,10 @@
-import{_ as r,c as n,j as t,a as e,G as l,a5 as a,B as d,o}from"./chunks/framework.Dr1kDDE8.js";const T1=JSON.parse('{"title":"Problem setup","description":"","frontmatter":{},"headers":[],"relativePath":"manual/setup.md","filePath":"manual/setup.md","lastUpdated":null}'),p={name:"manual/setup.md"},Q={class:"jldocstring custom-block",open:""},T={class:"jldocstring custom-block",open:""},m={class:"jldocstring custom-block",open:""},h={class:"jldocstring custom-block",open:""},c={class:"jldocstring custom-block",open:""},g={class:"jldocstring custom-block",open:""},k={class:"jldocstring custom-block",open:""},u={class:"jldocstring custom-block",open:""},b={class:"jldocstring custom-block",open:""},y={class:"jldocstring custom-block",open:""},f={class:"jldocstring custom-block",open:""},v={class:"jldocstring custom-block",open:""},E={class:"jldocstring custom-block",open:""},x={class:"jldocstring custom-block",open:""},w={class:"jldocstring custom-block",open:""},j={class:"jldocstring custom-block",open:""},H={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},M={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.339ex"},xmlns:"http://www.w3.org/2000/svg",width:"2.288ex",height:"1.339ex",role:"img",focusable:"false",viewBox:"0 -442 1011.4 592","aria-hidden":"true"},L={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},C={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.799ex"},xmlns:"http://www.w3.org/2000/svg",width:"5.227ex",height:"1.799ex",role:"img",focusable:"false",viewBox:"0 -442 2310.5 795.1","aria-hidden":"true"},N={class:"jldocstring custom-block",open:""},F={class:"MathJax",jax:"SVG",display:"true",style:{direction:"ltr",display:"block","text-align":"center",margin:"1em 0",position:"relative"}},Z={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-2.148ex"},xmlns:"http://www.w3.org/2000/svg",width:"50.685ex",height:"5.428ex",role:"img",focusable:"false",viewBox:"0 -1449.5 22402.7 2399","aria-hidden":"true"},S={class:"jldocstring custom-block",open:""},A={class:"jldocstring custom-block",open:""},D={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},I={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.09ex"},xmlns:"http://www.w3.org/2000/svg",width:"9.346ex",height:"2.004ex",role:"img",focusable:"false",viewBox:"0 -846 4131.1 886","aria-hidden":"true"},V={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},B={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.777ex"},xmlns:"http://www.w3.org/2000/svg",width:"20.12ex",height:"2.664ex",role:"img",focusable:"false",viewBox:"0 -833.9 8893.2 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1312.8","aria-hidden":"true"},q={class:"MathJax",jax:"SVG",display:"true",style:{direction:"ltr",display:"block","text-align":"center",margin:"1em 0",position:"relative"}},U={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-1.842ex"},xmlns:"http://www.w3.org/2000/svg",width:"40.13ex",height:"4.899ex",role:"img",focusable:"false",viewBox:"0 -1351.5 17737.3 2165.5","aria-hidden":"true"},X={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},K={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.471ex"},xmlns:"http://www.w3.org/2000/svg",width:"4.486ex",height:"1.471ex",role:"img",focusable:"false",viewBox:"0 -442 1982.9 650","aria-hidden":"true"},$={class:"jldocstring custom-block",open:""},W={class:"jldocstring custom-block",open:""},Y={class:"jldocstring custom-block",open:""},_={class:"jldocstring custom-block",open:""},t1={class:"jldocstring custom-block",open:""},s1={class:"jldocstring custom-block",open:""},e1={class:"jldocstring custom-block",open:""},a1={class:"jldocstring custom-block",open:""},i1={class:"jldocstring custom-block",open:""};function l1(n1,s,o1,r1,d1,p1){const i=d("Badge");return o(),n("div",null,[s[124]||(s[124]=t("h1",{id:"Problem-setup",tabindex:"-1"},[e("Problem setup "),t("a",{class:"header-anchor",href:"#Problem-setup","aria-label":'Permalink to "Problem setup {#Problem-setup}"'},"​")],-1)),s[125]||(s[125]=t("h2",{id:"Boundary-conditions",tabindex:"-1"},[e("Boundary conditions "),t("a",{class:"header-anchor",href:"#Boundary-conditions","aria-label":'Permalink to "Boundary conditions {#Boundary-conditions}"'},"​")],-1)),s[126]||(s[126]=t("p",null,"Each boundary has exactly one type of boundary conditions. For periodic boundary conditions, the opposite boundary must also be periodic.",-1)),t("details",Q,[t("summary",null,[s[0]||(s[0]=t("a",{id:"IncompressibleNavierStokes.AbstractBC",href:"#IncompressibleNavierStokes.AbstractBC"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.AbstractBC")],-1)),s[1]||(s[1]=e()),l(i,{type:"info",class:"jlObjectType jlType",text:"Type"})]),s[2]||(s[2]=a('
julia
abstract type AbstractBC
Boundary condition for one side of the domain.
Fields
source
',4))]),t("details",T,[t("summary",null,[s[3]||(s[3]=t("a",{id:"IncompressibleNavierStokes.DirichletBC",href:"#IncompressibleNavierStokes.DirichletBC"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.DirichletBC")],-1)),s[4]||(s[4]=e()),l(i,{type:"info",class:"jlObjectType jlType",text:"Type"})]),s[5]||(s[5]=a('
julia
struct DirichletBC{U} <: IncompressibleNavierStokes.AbstractBC
Dirichlet boundary conditions for the velocity, where u[1] = (x..., t) -> u1_BC up to u[d] = (x..., t) -> ud_BC, where d is the dimension.
When u is nothing, then the boundary conditions are no slip boundary conditions, where all velocity components are zero.
Fields
  • u: Boundary condition
source
',6))]),t("details",m,[t("summary",null,[s[6]||(s[6]=t("a",{id:"IncompressibleNavierStokes.PeriodicBC",href:"#IncompressibleNavierStokes.PeriodicBC"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.PeriodicBC")],-1)),s[7]||(s[7]=e()),l(i,{type:"info",class:"jlObjectType jlType",text:"Type"})]),s[8]||(s[8]=a('
julia
struct PeriodicBC <: IncompressibleNavierStokes.AbstractBC
Periodic boundary conditions. Must be periodic on both sides.
Fields
source
',4))]),t("details",h,[t("summary",null,[s[9]||(s[9]=t("a",{id:"IncompressibleNavierStokes.PressureBC",href:"#IncompressibleNavierStokes.PressureBC"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.PressureBC")],-1)),s[10]||(s[10]=e()),l(i,{type:"info",class:"jlObjectType jlType",text:"Type"})]),s[11]||(s[11]=a('
julia
struct PressureBC <: IncompressibleNavierStokes.AbstractBC
Pressure boundary conditions. The pressure is prescribed on the boundary (usually an "outlet"). The velocity has zero Neumann conditions.
Note: Currently, the pressure is prescribed with the constant value of zero on the entire boundary.
Fields
source
',5))]),t("details",c,[t("summary",null,[s[12]||(s[12]=t("a",{id:"IncompressibleNavierStokes.SymmetricBC",href:"#IncompressibleNavierStokes.SymmetricBC"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.SymmetricBC")],-1)),s[13]||(s[13]=e()),l(i,{type:"info",class:"jlObjectType jlType",text:"Type"})]),s[14]||(s[14]=a('
julia
struct SymmetricBC <: IncompressibleNavierStokes.AbstractBC
Symmetric boundary conditions. The parallel velocity and pressure is the same at each side of the boundary. The normal velocity is zero.
Fields
source
',4))]),t("details",g,[t("summary",null,[s[15]||(s[15]=t("a",{id:"IncompressibleNavierStokes.apply_bc_p!-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.apply_bc_p!-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.apply_bc_p!")],-1)),s[16]||(s[16]=e()),l(i,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[17]||(s[17]=a('
julia
apply_bc_p!(p, t, setup; kwargs...) -> Any
Apply pressure boundary conditions (in-place version).
source
',3))]),t("details",k,[t("summary",null,[s[18]||(s[18]=t("a",{id:"IncompressibleNavierStokes.apply_bc_p-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.apply_bc_p-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.apply_bc_p")],-1)),s[19]||(s[19]=e()),l(i,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[20]||(s[20]=a('
julia
apply_bc_p(p, t, setup; kwargs...) -> Any
Apply pressure boundary conditions (differentiable version).
source
',3))]),t("details",u,[t("summary",null,[s[21]||(s[21]=t("a",{id:"IncompressibleNavierStokes.apply_bc_temp!-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.apply_bc_temp!-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.apply_bc_temp!")],-1)),s[22]||(s[22]=e()),l(i,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[23]||(s[23]=a('
julia
apply_bc_temp!(temp, t, setup; kwargs...) -> Any
Apply temperature boundary conditions (in-place version).
source
',3))]),t("details",b,[t("summary",null,[s[24]||(s[24]=t("a",{id:"IncompressibleNavierStokes.apply_bc_temp-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.apply_bc_temp-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.apply_bc_temp")],-1)),s[25]||(s[25]=e()),l(i,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[26]||(s[26]=a('
julia
apply_bc_temp(temp, t, setup; kwargs...) -> Any
Apply temperature boundary conditions (differentiable version).
source
',3))]),t("details",y,[t("summary",null,[s[27]||(s[27]=t("a",{id:"IncompressibleNavierStokes.apply_bc_u!-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.apply_bc_u!-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.apply_bc_u!")],-1)),s[28]||(s[28]=e()),l(i,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[29]||(s[29]=a('
julia
apply_bc_u!(u, t, setup; kwargs...) -> Any
Apply velocity boundary conditions (in-place version).
source
',3))]),t("details",f,[t("summary",null,[s[30]||(s[30]=t("a",{id:"IncompressibleNavierStokes.apply_bc_u-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.apply_bc_u-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.apply_bc_u")],-1)),s[31]||(s[31]=e()),l(i,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[32]||(s[32]=a('
julia
apply_bc_u(u, t, setup; kwargs...) -> Any
Apply velocity boundary conditions (differentiable version).
source
',3))]),t("details",v,[t("summary",null,[s[33]||(s[33]=t("a",{id:"IncompressibleNavierStokes.boundary-NTuple{4, Any}",href:"#IncompressibleNavierStokes.boundary-NTuple{4, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.boundary")],-1)),s[34]||(s[34]=e()),l(i,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[35]||(s[35]=a('
julia
boundary(β, N, I, isright) -> Any
Get boundary indices of boundary layer normal to β. The CartesianIndices given by I should contain those of the inner DOFs, typically Ip or Iu[α]. The boundary layer is then just outside those.
source
',3))]),t("details",E,[t("summary",null,[s[36]||(s[36]=t("a",{id:"IncompressibleNavierStokes.offset_p",href:"#IncompressibleNavierStokes.offset_p"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.offset_p")],-1)),s[37]||(s[37]=e()),l(i,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[38]||(s[38]=a('
julia
offset_p(bc, isnormal, isright)
Number of non-DOF pressure components at boundary.
source
',3))]),t("details",x,[t("summary",null,[s[39]||(s[39]=t("a",{id:"IncompressibleNavierStokes.offset_u",href:"#IncompressibleNavierStokes.offset_u"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.offset_u")],-1)),s[40]||(s[40]=e()),l(i,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[41]||(s[41]=a('
julia
offset_u(bc, isnormal, isright)
Number of non-DOF velocity components at boundary. If isnormal, then the velocity is normal to the boundary, else parallel. If isright, it is at the end/right/rear/top boundary, otherwise beginning.
source
',3))]),s[127]||(s[127]=t("h2",{id:"grid",tabindex:"-1"},[e("Grid "),t("a",{class:"header-anchor",href:"#grid","aria-label":'Permalink to "Grid"'},"​")],-1)),t("details",w,[t("summary",null,[s[42]||(s[42]=t("a",{id:"IncompressibleNavierStokes.Dimension",href:"#IncompressibleNavierStokes.Dimension"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.Dimension")],-1)),s[43]||(s[43]=e()),l(i,{type:"info",class:"jlObjectType jlType",text:"Type"})]),s[44]||(s[44]=a(`
julia
struct Dimension{N}
Represent an N-dimensional space. Returns N when called.
julia
julia> d = Dimension(3)
+import{_ as r,c as n,j as t,a as e,G as l,a5 as a,B as d,o}from"./chunks/framework.Dr1kDDE8.js";const T1=JSON.parse('{"title":"Problem setup","description":"","frontmatter":{},"headers":[],"relativePath":"manual/setup.md","filePath":"manual/setup.md","lastUpdated":null}'),p={name:"manual/setup.md"},Q={class:"jldocstring custom-block",open:""},T={class:"jldocstring custom-block",open:""},m={class:"jldocstring custom-block",open:""},h={class:"jldocstring custom-block",open:""},c={class:"jldocstring custom-block",open:""},g={class:"jldocstring custom-block",open:""},k={class:"jldocstring custom-block",open:""},u={class:"jldocstring custom-block",open:""},b={class:"jldocstring custom-block",open:""},y={class:"jldocstring custom-block",open:""},f={class:"jldocstring custom-block",open:""},v={class:"jldocstring custom-block",open:""},E={class:"jldocstring custom-block",open:""},x={class:"jldocstring custom-block",open:""},w={class:"jldocstring custom-block",open:""},j={class:"jldocstring custom-block",open:""},H={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},M={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.339ex"},xmlns:"http://www.w3.org/2000/svg",width:"2.288ex",height:"1.339ex",role:"img",focusable:"false",viewBox:"0 -442 1011.4 592","aria-hidden":"true"},L={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},C={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.799ex"},xmlns:"http://www.w3.org/2000/svg",width:"5.227ex",height:"1.799ex",role:"img",focusable:"false",viewBox:"0 -442 2310.5 795.1","aria-hidden":"true"},N={class:"jldocstring custom-block",open:""},F={class:"MathJax",jax:"SVG",display:"true",style:{direction:"ltr",display:"block","text-align":"center",margin:"1em 0",position:"relative"}},Z={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-2.148ex"},xmlns:"http://www.w3.org/2000/svg",width:"50.685ex",height:"5.428ex",role:"img",focusable:"false",viewBox:"0 -1449.5 22402.7 2399","aria-hidden":"true"},S={class:"jldocstring custom-block",open:""},A={class:"jldocstring custom-block",open:""},D={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},I={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.09ex"},xmlns:"http://www.w3.org/2000/svg",width:"9.346ex",height:"2.004ex",role:"img",focusable:"false",viewBox:"0 -846 4131.1 886","aria-hidden":"true"},V={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},B={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.777ex"},xmlns:"http://www.w3.org/2000/svg",width:"20.12ex",height:"2.664ex",role:"img",focusable:"false",viewBox:"0 -833.9 8893.2 1177.3","aria-hidden":"true"},O={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},P={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.439ex"},xmlns:"http://www.w3.org/2000/svg",width:"12.556ex",height:"1.984ex",role:"img",focusable:"false",viewBox:"0 -683 5549.6 877","aria-hidden":"true"},z={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},G={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.339ex"},xmlns:"http://www.w3.org/2000/svg",width:"6.89ex",height:"1.91ex",role:"img",focusable:"false",viewBox:"0 -694 3045.5 844","aria-hidden":"true"},R={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},J={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-1.004ex"},xmlns:"http://www.w3.org/2000/svg",width:"15.941ex",height:"2.97ex",role:"img",focusable:"false",viewBox:"0 -868.9 7046 1312.8","aria-hidden":"true"},q={class:"MathJax",jax:"SVG",display:"true",style:{direction:"ltr",display:"block","text-align":"center",margin:"1em 0",position:"relative"}},U={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-1.842ex"},xmlns:"http://www.w3.org/2000/svg",width:"40.13ex",height:"4.899ex",role:"img",focusable:"false",viewBox:"0 -1351.5 17737.3 2165.5","aria-hidden":"true"},X={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},K={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.471ex"},xmlns:"http://www.w3.org/2000/svg",width:"4.486ex",height:"1.471ex",role:"img",focusable:"false",viewBox:"0 -442 1982.9 650","aria-hidden":"true"},$={class:"jldocstring custom-block",open:""},W={class:"jldocstring custom-block",open:""},Y={class:"jldocstring custom-block",open:""},_={class:"jldocstring custom-block",open:""},t1={class:"jldocstring custom-block",open:""},s1={class:"jldocstring custom-block",open:""},e1={class:"jldocstring custom-block",open:""},a1={class:"jldocstring custom-block",open:""},i1={class:"jldocstring custom-block",open:""};function l1(n1,s,o1,r1,d1,p1){const i=d("Badge");return o(),n("div",null,[s[124]||(s[124]=t("h1",{id:"Problem-setup",tabindex:"-1"},[e("Problem setup "),t("a",{class:"header-anchor",href:"#Problem-setup","aria-label":'Permalink to "Problem setup {#Problem-setup}"'},"​")],-1)),s[125]||(s[125]=t("h2",{id:"Boundary-conditions",tabindex:"-1"},[e("Boundary conditions "),t("a",{class:"header-anchor",href:"#Boundary-conditions","aria-label":'Permalink to "Boundary conditions {#Boundary-conditions}"'},"​")],-1)),s[126]||(s[126]=t("p",null,"Each boundary has exactly one type of boundary conditions. For periodic boundary conditions, the opposite boundary must also be periodic.",-1)),t("details",Q,[t("summary",null,[s[0]||(s[0]=t("a",{id:"IncompressibleNavierStokes.AbstractBC",href:"#IncompressibleNavierStokes.AbstractBC"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.AbstractBC")],-1)),s[1]||(s[1]=e()),l(i,{type:"info",class:"jlObjectType jlType",text:"Type"})]),s[2]||(s[2]=a('
julia
abstract type AbstractBC
Boundary condition for one side of the domain.
Fields
source
',4))]),t("details",T,[t("summary",null,[s[3]||(s[3]=t("a",{id:"IncompressibleNavierStokes.DirichletBC",href:"#IncompressibleNavierStokes.DirichletBC"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.DirichletBC")],-1)),s[4]||(s[4]=e()),l(i,{type:"info",class:"jlObjectType jlType",text:"Type"})]),s[5]||(s[5]=a('
julia
struct DirichletBC{U} <: IncompressibleNavierStokes.AbstractBC
Dirichlet boundary conditions for the velocity, where u[1] = (x..., t) -> u1_BC up to u[d] = (x..., t) -> ud_BC, where d is the dimension.
When u is nothing, then the boundary conditions are no slip boundary conditions, where all velocity components are zero.
Fields
  • u: Boundary condition
source
',6))]),t("details",m,[t("summary",null,[s[6]||(s[6]=t("a",{id:"IncompressibleNavierStokes.PeriodicBC",href:"#IncompressibleNavierStokes.PeriodicBC"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.PeriodicBC")],-1)),s[7]||(s[7]=e()),l(i,{type:"info",class:"jlObjectType jlType",text:"Type"})]),s[8]||(s[8]=a('
julia
struct PeriodicBC <: IncompressibleNavierStokes.AbstractBC
Periodic boundary conditions. Must be periodic on both sides.
Fields
source
',4))]),t("details",h,[t("summary",null,[s[9]||(s[9]=t("a",{id:"IncompressibleNavierStokes.PressureBC",href:"#IncompressibleNavierStokes.PressureBC"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.PressureBC")],-1)),s[10]||(s[10]=e()),l(i,{type:"info",class:"jlObjectType jlType",text:"Type"})]),s[11]||(s[11]=a('
julia
struct PressureBC <: IncompressibleNavierStokes.AbstractBC
Pressure boundary conditions. The pressure is prescribed on the boundary (usually an "outlet"). The velocity has zero Neumann conditions.
Note: Currently, the pressure is prescribed with the constant value of zero on the entire boundary.
Fields
source
',5))]),t("details",c,[t("summary",null,[s[12]||(s[12]=t("a",{id:"IncompressibleNavierStokes.SymmetricBC",href:"#IncompressibleNavierStokes.SymmetricBC"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.SymmetricBC")],-1)),s[13]||(s[13]=e()),l(i,{type:"info",class:"jlObjectType jlType",text:"Type"})]),s[14]||(s[14]=a('
julia
struct SymmetricBC <: IncompressibleNavierStokes.AbstractBC
Symmetric boundary conditions. The parallel velocity and pressure is the same at each side of the boundary. The normal velocity is zero.
Fields
source
',4))]),t("details",g,[t("summary",null,[s[15]||(s[15]=t("a",{id:"IncompressibleNavierStokes.apply_bc_p!-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.apply_bc_p!-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.apply_bc_p!")],-1)),s[16]||(s[16]=e()),l(i,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[17]||(s[17]=a('
julia
apply_bc_p!(p, t, setup; kwargs...) -> Any
Apply pressure boundary conditions (in-place version).
source
',3))]),t("details",k,[t("summary",null,[s[18]||(s[18]=t("a",{id:"IncompressibleNavierStokes.apply_bc_p-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.apply_bc_p-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.apply_bc_p")],-1)),s[19]||(s[19]=e()),l(i,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[20]||(s[20]=a('
julia
apply_bc_p(p, t, setup; kwargs...) -> Any
Apply pressure boundary conditions (differentiable version).
source
',3))]),t("details",u,[t("summary",null,[s[21]||(s[21]=t("a",{id:"IncompressibleNavierStokes.apply_bc_temp!-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.apply_bc_temp!-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.apply_bc_temp!")],-1)),s[22]||(s[22]=e()),l(i,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[23]||(s[23]=a('
julia
apply_bc_temp!(temp, t, setup; kwargs...) -> Any
Apply temperature boundary conditions (in-place version).
source
',3))]),t("details",b,[t("summary",null,[s[24]||(s[24]=t("a",{id:"IncompressibleNavierStokes.apply_bc_temp-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.apply_bc_temp-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.apply_bc_temp")],-1)),s[25]||(s[25]=e()),l(i,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[26]||(s[26]=a('
julia
apply_bc_temp(temp, t, setup; kwargs...) -> Any
Apply temperature boundary conditions (differentiable version).
source
',3))]),t("details",y,[t("summary",null,[s[27]||(s[27]=t("a",{id:"IncompressibleNavierStokes.apply_bc_u!-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.apply_bc_u!-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.apply_bc_u!")],-1)),s[28]||(s[28]=e()),l(i,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[29]||(s[29]=a('
julia
apply_bc_u!(u, t, setup; kwargs...) -> Any
Apply velocity boundary conditions (in-place version).
source
',3))]),t("details",f,[t("summary",null,[s[30]||(s[30]=t("a",{id:"IncompressibleNavierStokes.apply_bc_u-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.apply_bc_u-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.apply_bc_u")],-1)),s[31]||(s[31]=e()),l(i,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[32]||(s[32]=a('
julia
apply_bc_u(u, t, setup; kwargs...) -> Any
Apply velocity boundary conditions (differentiable version).
source
',3))]),t("details",v,[t("summary",null,[s[33]||(s[33]=t("a",{id:"IncompressibleNavierStokes.boundary-NTuple{4, Any}",href:"#IncompressibleNavierStokes.boundary-NTuple{4, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.boundary")],-1)),s[34]||(s[34]=e()),l(i,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[35]||(s[35]=a('
julia
boundary(β, N, I, isright) -> Any
Get boundary indices of boundary layer normal to β. The CartesianIndices given by I should contain those of the inner DOFs, typically Ip or Iu[α]. The boundary layer is then just outside those.
source
',3))]),t("details",E,[t("summary",null,[s[36]||(s[36]=t("a",{id:"IncompressibleNavierStokes.offset_p",href:"#IncompressibleNavierStokes.offset_p"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.offset_p")],-1)),s[37]||(s[37]=e()),l(i,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[38]||(s[38]=a('
julia
offset_p(bc, isnormal, isright)
Number of non-DOF pressure components at boundary.
source
',3))]),t("details",x,[t("summary",null,[s[39]||(s[39]=t("a",{id:"IncompressibleNavierStokes.offset_u",href:"#IncompressibleNavierStokes.offset_u"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.offset_u")],-1)),s[40]||(s[40]=e()),l(i,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[41]||(s[41]=a('
julia
offset_u(bc, isnormal, isright)
Number of non-DOF velocity components at boundary. If isnormal, then the velocity is normal to the boundary, else parallel. If isright, it is at the end/right/rear/top boundary, otherwise beginning.
source
',3))]),s[127]||(s[127]=t("h2",{id:"grid",tabindex:"-1"},[e("Grid "),t("a",{class:"header-anchor",href:"#grid","aria-label":'Permalink to "Grid"'},"​")],-1)),t("details",w,[t("summary",null,[s[42]||(s[42]=t("a",{id:"IncompressibleNavierStokes.Dimension",href:"#IncompressibleNavierStokes.Dimension"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.Dimension")],-1)),s[43]||(s[43]=e()),l(i,{type:"info",class:"jlObjectType jlType",text:"Type"})]),s[44]||(s[44]=a(`
julia
struct Dimension{N}
Represent an N-dimensional space. Returns N when called.
julia
julia> d = Dimension(3)
 Dimension{3}()
 
 julia> d()
-3
Fields
source
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julia
Grid(; x, boundary_conditions, backend)
Create useful quantities for Cartesian box mesh ``x[1] \\times \\dots \\times x[d]with boundary conditionsboundary_conditions. Return a named tuple ([α]` denotes a tuple index) with the following fields:
',2)),t("ul",null,[s[54]||(s[54]=a("
  • N[α]: Number of finite volumes in direction β, including ghost volumes
  • Nu[α][β]: Number of u[α] velocity DOFs in direction β
  • Np[α]: Number of pressure DOFs in direction α
  • Iu[α]: Cartesian index range of u[α] velocity DOFs
  • Ip: Cartesian index range of pressure DOFs
  • xlims[α]: Tuple containing the limits of the physical domain (not grid) in the direction α
  • x[α]: α-coordinates of all volume boundaries, including the left point of the first ghost volume
  • xu[α][β]: β-coordinates of u[α] velocity points
  • xp[α]: α-coordinates of pressure points
  • Δ[α]: All volume widths in direction α
  • Δu[α]: Distance between pressure points in direction α
    • ",11)),t("li",null,[t("p",null,[s[51]||(s[51]=t("code",null,"A[α][β]",-1)),s[52]||(s[52]=e(": Interpolation weights from α-face centers ")),t("mjx-container",H,[(o(),n("svg",M,s[47]||(s[47]=[a('',1)]))),s[48]||(s[48]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"x"),t("mi",null,"I")])])],-1))]),s[53]||(s[53]=e(" to ")),t("mjx-container",L,[(o(),n("svg",C,s[49]||(s[49]=[a('',1)]))),s[50]||(s[50]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"x"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",null,"I"),t("mo",null,"±"),t("msub",null,[t("mi",null,"h"),t("mi",null,"β")])])])])],-1))])])])]),s[56]||(s[56]=t("p",null,"Note that the memory footprint of the redundant 1D-arrays above is negligible compared to the memory footprint of the 2D/3D-fields used in the code.",-1)),s[57]||(s[57]=t("p",null,[t("a",{href:"https://github.com/agdestein/IncompressibleNavierStokes.jl/blob/03acafa3566742b1f37d722d2207f5dc21d031c0/src/grid.jl#L79",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",N,[t("summary",null,[s[58]||(s[58]=t("a",{id:"IncompressibleNavierStokes.cosine_grid-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.cosine_grid-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.cosine_grid")],-1)),s[59]||(s[59]=e()),l(i,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[62]||(s[62]=a('
    julia
    cosine_grid(a, b, N) -> Any
    Create a nonuniform grid of N + 1 points from a to b using a cosine profile, i.e.
    ',2)),t("mjx-container",F,[(o(),n("svg",Z,s[60]||(s[60]=[a('',1)]))),s[61]||(s[61]=t("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[t("msub",null,[t("mi",null,"x"),t("mi",null,"i")]),t("mo",null,"="),t("mi",null,"a"),t("mo",null,"+"),t("mfrac",null,[t("mn",null,"1"),t("mn",null,"2")]),t("mrow",{"data-mjx-texclass":"INNER"},[t("mo",{"data-mjx-texclass":"OPEN"},"("),t("mn",null,"1"),t("mo",null,"−"),t("mi",null,"cos"),t("mo",{"data-mjx-texclass":"NONE"},"⁡"),t("mrow",{"data-mjx-texclass":"INNER"},[t("mo",{"data-mjx-texclass":"OPEN"},"("),t("mi",null,"π"),t("mfrac",null,[t("mi",null,"i"),t("mi",null,"n")]),t("mo",{"data-mjx-texclass":"CLOSE"},")")]),t("mo",{"data-mjx-texclass":"CLOSE"},")")]),t("mo",{stretchy:"false"},"("),t("mi",null,"b"),t("mo",null,"−"),t("mi",null,"a"),t("mo",{stretchy:"false"},")"),t("mo",null,","),t("mstyle",{scriptlevel:"0"},[t("mspace",{width:"1em"})]),t("mi",null,"i"),t("mo",null,"="),t("mn",null,"0"),t("mo",null,","),t("mo",null,"…"),t("mo",null,","),t("mi",null,"N")])],-1))]),s[63]||(s[63]=t("p",null,[e("See also "),t("a",{href:"/IncompressibleNavierStokes.jl/dev/manual/setup#IncompressibleNavierStokes.stretched_grid"},[t("code",null,"stretched_grid")]),e(".")],-1)),s[64]||(s[64]=t("p",null,[t("a",{href:"https://github.com/agdestein/IncompressibleNavierStokes.jl/blob/03acafa3566742b1f37d722d2207f5dc21d031c0/src/grid.jl#L28",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",S,[t("summary",null,[s[65]||(s[65]=t("a",{id:"IncompressibleNavierStokes.max_size-Tuple{Any}",href:"#IncompressibleNavierStokes.max_size-Tuple{Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.max_size")],-1)),s[66]||(s[66]=e()),l(i,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[67]||(s[67]=a('
    julia
    max_size(grid) -> Any
    Get size of the largest grid element.
    source
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    julia
    stretched_grid(a, b, N) -> Any
-stretched_grid(a, b, N, s) -> Any
    `,1)),t("p",null,[s[80]||(s[80]=a("Create a nonuniform grid of N + 1 points from a to b with a stretch factor of s. If s = 1, return a uniform spacing from a to b. 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    julia
    tanh_grid(a, b, N) -> Any
-tanh_grid(a, b, N, γ) -> Any
    Create a nonuniform grid of N + 1 points from a to b, as proposed by Trias et al. [10].
    source
    `,3))]),s[128]||(s[128]=t("h2",{id:"setup",tabindex:"-1"},[e("Setup "),t("a",{class:"header-anchor",href:"#setup","aria-label":'Permalink to "Setup"'},"​")],-1)),t("details",W,[t("summary",null,[s[100]||(s[100]=t("a",{id:"KernelAbstractions.CPU",href:"#KernelAbstractions.CPU"},[t("span",{class:"jlbinding"},"KernelAbstractions.CPU")],-1)),s[101]||(s[101]=e()),l(i,{type:"info",class:"jlObjectType jlType",text:"Type"})]),s[102]||(s[102]=a('
    julia
    struct CPU <: KernelAbstractions.Backend
    CPU(; static=false)
    Instantiate a CPU (multi-threaded) backend.
    Options:
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    • static
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    julia
    Setup(
+3
    Fields
    source
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    julia
    Grid(; x, boundary_conditions, backend)
    Create useful quantities for Cartesian box mesh ``x[1] \\times \\dots \\times x[d]with boundary conditionsboundary_conditions. Return a named tuple ([α]` denotes a tuple index) with the following fields:
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  • N[α]: Number of finite volumes in direction β, including ghost volumes
  • Nu[α][β]: Number of u[α] velocity DOFs in direction β
  • Np[α]: Number of pressure DOFs in direction α
  • Iu[α]: Cartesian index range of u[α] velocity DOFs
  • Ip: Cartesian index range of pressure DOFs
  • xlims[α]: Tuple containing the limits of the physical domain (not grid) in the direction α
  • x[α]: α-coordinates of all volume boundaries, including the left point of the first ghost volume
  • xu[α][β]: β-coordinates of u[α] velocity points
  • xp[α]: α-coordinates of pressure points
  • Δ[α]: All volume widths in direction α
  • Δu[α]: Distance between pressure points in direction α
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    Get size of the largest grid element.
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    julia
    tanh_grid(a, b, N) -> Any
+tanh_grid(a, b, N, γ) -> Any
    Create a nonuniform grid of N + 1 points from a to b, as proposed by Trias et al. [10].
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    Instantiate a CPU (multi-threaded) backend.
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 ;
     x,
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@@ -16,7 +16,7 @@ import{_ as r,c as n,j as t,a as e,G as l,a5 as a,B as d,o}from"./chunks/framewo
     workgroupsize,
     temperature,
     Re
-)
    Create problem setup (stored in a named tuple).
    source
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    temperature_equation(
+)
    Create problem setup (stored in a named tuple).
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@@ -25,7 +25,7 @@ import{_ as r,c as n,j as t,a as e,G as l,a5 as a,B as d,o}from"./chunks/framewo
     boundary_conditions,
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-)
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    julia
    random_field(setup; ...) -> Any
-random_field(setup, t; A, kp, psolver, rng) -> Any
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    julia
    random_field(setup; ...) -> Any
+random_field(setup, t; A, kp, psolver, rng) -> Any
    Create random field, as in [11].
    • A: Eddy amplitude scaling
    • kp: Peak energy wavenumber
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    scalarfield(setup) -> Any
    Create empty scalar field.
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    julia
    temperaturefield(setup, tempfunc) -> Any
+temperaturefield(setup, tempfunc, t) -> Any
    Create temperature field from function with boundary conditions at time t.
    source
    `,3))]),t("details",a1,[t("summary",null,[s[118]||(s[118]=t("a",{id:"IncompressibleNavierStokes.vectorfield-Tuple{Any}",href:"#IncompressibleNavierStokes.vectorfield-Tuple{Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.vectorfield")],-1)),s[119]||(s[119]=e()),l(i,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[120]||(s[120]=a('
    julia
    vectorfield(setup) -> Any
    Create empty vector field.
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    ',3))]),t("details",i1,[t("summary",null,[s[121]||(s[121]=t("a",{id:"IncompressibleNavierStokes.velocityfield",href:"#IncompressibleNavierStokes.velocityfield"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.velocityfield")],-1)),s[122]||(s[122]=e()),l(i,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[123]||(s[123]=a(`
    julia
    velocityfield(setup, ufunc; ...) -> Any
+velocityfield(setup, ufunc, t; psolver, doproject) -> Any
    Create divergence free velocity field u with boundary conditions at time t. The initial conditions of u[α] are specified by the function ufunc(α, x...).
    source
    `,3))])])}const m1=r(p,[["render",l1]]);export{T1 as __pageData,m1 as default};
diff --git a/dev/assets/manual_setup.md.w6x6ynsm.lean.js b/dev/assets/manual_setup.md.B7b6UkIH.lean.js
similarity index 97%
rename from dev/assets/manual_setup.md.w6x6ynsm.lean.js
rename to dev/assets/manual_setup.md.B7b6UkIH.lean.js
index b1bac10f5..6084c86b4 100644
--- a/dev/assets/manual_setup.md.w6x6ynsm.lean.js
+++ b/dev/assets/manual_setup.md.B7b6UkIH.lean.js
@@ -1,10 +1,10 @@
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custom-block",open:""},e1={class:"jldocstring custom-block",open:""},a1={class:"jldocstring custom-block",open:""},i1={class:"jldocstring custom-block",open:""};function l1(n1,s,o1,r1,d1,p1){const i=d("Badge");return o(),n("div",null,[s[124]||(s[124]=t("h1",{id:"Problem-setup",tabindex:"-1"},[e("Problem setup "),t("a",{class:"header-anchor",href:"#Problem-setup","aria-label":'Permalink to "Problem setup {#Problem-setup}"'},"​")],-1)),s[125]||(s[125]=t("h2",{id:"Boundary-conditions",tabindex:"-1"},[e("Boundary conditions "),t("a",{class:"header-anchor",href:"#Boundary-conditions","aria-label":'Permalink to "Boundary conditions {#Boundary-conditions}"'},"​")],-1)),s[126]||(s[126]=t("p",null,"Each boundary has exactly one type of boundary conditions. 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    julia
    abstract type AbstractBC
    Boundary condition for one side of the domain.
    Fields
    source
    ',4))]),t("details",T,[t("summary",null,[s[3]||(s[3]=t("a",{id:"IncompressibleNavierStokes.DirichletBC",href:"#IncompressibleNavierStokes.DirichletBC"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.DirichletBC")],-1)),s[4]||(s[4]=e()),l(i,{type:"info",class:"jlObjectType jlType",text:"Type"})]),s[5]||(s[5]=a('
    julia
    struct DirichletBC{U} <: IncompressibleNavierStokes.AbstractBC
    Dirichlet boundary conditions for the velocity, where u[1] = (x..., t) -> u1_BC up to u[d] = (x..., t) -> ud_BC, where d is the dimension.
    When u is nothing, then the boundary conditions are no slip boundary conditions, where all velocity components are zero.
    Fields
    • u: Boundary condition
    source
    ',6))]),t("details",m,[t("summary",null,[s[6]||(s[6]=t("a",{id:"IncompressibleNavierStokes.PeriodicBC",href:"#IncompressibleNavierStokes.PeriodicBC"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.PeriodicBC")],-1)),s[7]||(s[7]=e()),l(i,{type:"info",class:"jlObjectType jlType",text:"Type"})]),s[8]||(s[8]=a('
    julia
    struct PeriodicBC <: IncompressibleNavierStokes.AbstractBC
    Periodic boundary conditions. Must be periodic on both sides.
    Fields
    source
    ',4))]),t("details",h,[t("summary",null,[s[9]||(s[9]=t("a",{id:"IncompressibleNavierStokes.PressureBC",href:"#IncompressibleNavierStokes.PressureBC"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.PressureBC")],-1)),s[10]||(s[10]=e()),l(i,{type:"info",class:"jlObjectType jlType",text:"Type"})]),s[11]||(s[11]=a('
    julia
    struct PressureBC <: IncompressibleNavierStokes.AbstractBC
    Pressure boundary conditions. The pressure is prescribed on the boundary (usually an "outlet"). The velocity has zero Neumann conditions.
    Note: Currently, the pressure is prescribed with the constant value of zero on the entire boundary.
    Fields
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    ',5))]),t("details",c,[t("summary",null,[s[12]||(s[12]=t("a",{id:"IncompressibleNavierStokes.SymmetricBC",href:"#IncompressibleNavierStokes.SymmetricBC"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.SymmetricBC")],-1)),s[13]||(s[13]=e()),l(i,{type:"info",class:"jlObjectType jlType",text:"Type"})]),s[14]||(s[14]=a('
    julia
    struct SymmetricBC <: IncompressibleNavierStokes.AbstractBC
    Symmetric boundary conditions. The parallel velocity and pressure is the same at each side of the boundary. The normal velocity is zero.
    Fields
    source
    ',4))]),t("details",g,[t("summary",null,[s[15]||(s[15]=t("a",{id:"IncompressibleNavierStokes.apply_bc_p!-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.apply_bc_p!-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.apply_bc_p!")],-1)),s[16]||(s[16]=e()),l(i,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[17]||(s[17]=a('
    julia
    apply_bc_p!(p, t, setup; kwargs...) -> Any
    Apply pressure boundary conditions (in-place version).
    source
    ',3))]),t("details",k,[t("summary",null,[s[18]||(s[18]=t("a",{id:"IncompressibleNavierStokes.apply_bc_p-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.apply_bc_p-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.apply_bc_p")],-1)),s[19]||(s[19]=e()),l(i,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[20]||(s[20]=a('
    julia
    apply_bc_p(p, t, setup; kwargs...) -> Any
    Apply pressure boundary conditions (differentiable version).
    source
    ',3))]),t("details",u,[t("summary",null,[s[21]||(s[21]=t("a",{id:"IncompressibleNavierStokes.apply_bc_temp!-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.apply_bc_temp!-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.apply_bc_temp!")],-1)),s[22]||(s[22]=e()),l(i,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[23]||(s[23]=a('
    julia
    apply_bc_temp!(temp, t, setup; kwargs...) -> Any
    Apply temperature boundary conditions (in-place version).
    source
    ',3))]),t("details",b,[t("summary",null,[s[24]||(s[24]=t("a",{id:"IncompressibleNavierStokes.apply_bc_temp-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.apply_bc_temp-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.apply_bc_temp")],-1)),s[25]||(s[25]=e()),l(i,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[26]||(s[26]=a('
    julia
    apply_bc_temp(temp, t, setup; kwargs...) -> Any
    Apply temperature boundary conditions (differentiable version).
    source
    ',3))]),t("details",y,[t("summary",null,[s[27]||(s[27]=t("a",{id:"IncompressibleNavierStokes.apply_bc_u!-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.apply_bc_u!-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.apply_bc_u!")],-1)),s[28]||(s[28]=e()),l(i,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[29]||(s[29]=a('
    julia
    apply_bc_u!(u, t, setup; kwargs...) -> Any
    Apply velocity boundary conditions (in-place version).
    source
    ',3))]),t("details",f,[t("summary",null,[s[30]||(s[30]=t("a",{id:"IncompressibleNavierStokes.apply_bc_u-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.apply_bc_u-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.apply_bc_u")],-1)),s[31]||(s[31]=e()),l(i,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[32]||(s[32]=a('
    julia
    apply_bc_u(u, t, setup; kwargs...) -> Any
    Apply velocity boundary conditions (differentiable version).
    source
    ',3))]),t("details",v,[t("summary",null,[s[33]||(s[33]=t("a",{id:"IncompressibleNavierStokes.boundary-NTuple{4, Any}",href:"#IncompressibleNavierStokes.boundary-NTuple{4, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.boundary")],-1)),s[34]||(s[34]=e()),l(i,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[35]||(s[35]=a('
    julia
    boundary(β, N, I, isright) -> Any
    Get boundary indices of boundary layer normal to β. The CartesianIndices given by I should contain those of the inner DOFs, typically Ip or Iu[α]. The boundary layer is then just outside those.
    source
    ',3))]),t("details",E,[t("summary",null,[s[36]||(s[36]=t("a",{id:"IncompressibleNavierStokes.offset_p",href:"#IncompressibleNavierStokes.offset_p"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.offset_p")],-1)),s[37]||(s[37]=e()),l(i,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[38]||(s[38]=a('
    julia
    offset_p(bc, isnormal, isright)
    Number of non-DOF pressure components at boundary.
    source
    ',3))]),t("details",x,[t("summary",null,[s[39]||(s[39]=t("a",{id:"IncompressibleNavierStokes.offset_u",href:"#IncompressibleNavierStokes.offset_u"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.offset_u")],-1)),s[40]||(s[40]=e()),l(i,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[41]||(s[41]=a('
    julia
    offset_u(bc, isnormal, isright)
    Number of non-DOF velocity components at boundary. If isnormal, then the velocity is normal to the boundary, else parallel. If isright, it is at the end/right/rear/top boundary, otherwise beginning.
    source
    ',3))]),s[127]||(s[127]=t("h2",{id:"grid",tabindex:"-1"},[e("Grid "),t("a",{class:"header-anchor",href:"#grid","aria-label":'Permalink to "Grid"'},"​")],-1)),t("details",w,[t("summary",null,[s[42]||(s[42]=t("a",{id:"IncompressibleNavierStokes.Dimension",href:"#IncompressibleNavierStokes.Dimension"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.Dimension")],-1)),s[43]||(s[43]=e()),l(i,{type:"info",class:"jlObjectType jlType",text:"Type"})]),s[44]||(s[44]=a(`
    julia
    struct Dimension{N}
    Represent an N-dimensional space. Returns N when called.
    julia
    julia> d = Dimension(3)
+import{_ as r,c as n,j as t,a as e,G as l,a5 as a,B as d,o}from"./chunks/framework.Dr1kDDE8.js";const T1=JSON.parse('{"title":"Problem setup","description":"","frontmatter":{},"headers":[],"relativePath":"manual/setup.md","filePath":"manual/setup.md","lastUpdated":null}'),p={name:"manual/setup.md"},Q={class:"jldocstring custom-block",open:""},T={class:"jldocstring custom-block",open:""},m={class:"jldocstring custom-block",open:""},h={class:"jldocstring custom-block",open:""},c={class:"jldocstring custom-block",open:""},g={class:"jldocstring custom-block",open:""},k={class:"jldocstring custom-block",open:""},u={class:"jldocstring custom-block",open:""},b={class:"jldocstring custom-block",open:""},y={class:"jldocstring custom-block",open:""},f={class:"jldocstring custom-block",open:""},v={class:"jldocstring custom-block",open:""},E={class:"jldocstring custom-block",open:""},x={class:"jldocstring custom-block",open:""},w={class:"jldocstring custom-block",open:""},j={class:"jldocstring 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    julia
    abstract type AbstractBC
    Boundary condition for one side of the domain.
    Fields
    source
    ',4))]),t("details",T,[t("summary",null,[s[3]||(s[3]=t("a",{id:"IncompressibleNavierStokes.DirichletBC",href:"#IncompressibleNavierStokes.DirichletBC"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.DirichletBC")],-1)),s[4]||(s[4]=e()),l(i,{type:"info",class:"jlObjectType jlType",text:"Type"})]),s[5]||(s[5]=a('
    julia
    struct DirichletBC{U} <: IncompressibleNavierStokes.AbstractBC
    Dirichlet boundary conditions for the velocity, where u[1] = (x..., t) -> u1_BC up to u[d] = (x..., t) -> ud_BC, where d is the dimension.
    When u is nothing, then the boundary conditions are no slip boundary conditions, where all velocity components are zero.
    Fields
    • u: Boundary condition
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    julia
    struct PeriodicBC <: IncompressibleNavierStokes.AbstractBC
    Periodic boundary conditions. Must be periodic on both sides.
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    julia
    struct PressureBC <: IncompressibleNavierStokes.AbstractBC
    Pressure boundary conditions. The pressure is prescribed on the boundary (usually an "outlet"). The velocity has zero Neumann conditions.
    Note: Currently, the pressure is prescribed with the constant value of zero on the entire boundary.
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    julia
    struct SymmetricBC <: IncompressibleNavierStokes.AbstractBC
    Symmetric boundary conditions. The parallel velocity and pressure is the same at each side of the boundary. The normal velocity is zero.
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    julia
    apply_bc_p!(p, t, setup; kwargs...) -> Any
    Apply pressure boundary conditions (in-place version).
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    julia
    apply_bc_p(p, t, setup; kwargs...) -> Any
    Apply pressure boundary conditions (differentiable version).
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    julia
    apply_bc_temp!(temp, t, setup; kwargs...) -> Any
    Apply temperature boundary conditions (in-place version).
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    apply_bc_temp(temp, t, setup; kwargs...) -> Any
    Apply temperature boundary conditions (differentiable version).
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    julia
    apply_bc_u!(u, t, setup; kwargs...) -> Any
    Apply velocity boundary conditions (in-place version).
    source
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    julia
    apply_bc_u(u, t, setup; kwargs...) -> Any
    Apply velocity boundary conditions (differentiable version).
    source
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    julia
    boundary(β, N, I, isright) -> Any
    Get boundary indices of boundary layer normal to β. The CartesianIndices given by I should contain those of the inner DOFs, typically Ip or Iu[α]. The boundary layer is then just outside those.
    source
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    julia
    offset_p(bc, isnormal, isright)
    Number of non-DOF pressure components at boundary.
    source
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    julia
    offset_u(bc, isnormal, isright)
    Number of non-DOF velocity components at boundary. If isnormal, then the velocity is normal to the boundary, else parallel. If isright, it is at the end/right/rear/top boundary, otherwise beginning.
    source
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    julia
    struct Dimension{N}
    Represent an N-dimensional space. Returns N when called.
    julia
    julia> d = Dimension(3)
 Dimension{3}()
 
 julia> d()
-3
    Fields
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    julia
    Grid(; x, boundary_conditions, backend)
    Create useful quantities for Cartesian box mesh ``x[1] \\times \\dots \\times x[d]with boundary conditionsboundary_conditions. Return a named tuple ([α]` denotes a tuple index) with the following fields:
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  • N[α]: Number of finite volumes in direction β, including ghost volumes
  • Nu[α][β]: Number of u[α] velocity DOFs in direction β
  • Np[α]: Number of pressure DOFs in direction α
  • Iu[α]: Cartesian index range of u[α] velocity DOFs
  • Ip: Cartesian index range of pressure DOFs
  • xlims[α]: Tuple containing the limits of the physical domain (not grid) in the direction α
  • x[α]: α-coordinates of all volume boundaries, including the left point of the first ghost volume
  • xu[α][β]: β-coordinates of u[α] velocity points
  • xp[α]: α-coordinates of pressure points
  • Δ[α]: All volume widths in direction α
  • Δu[α]: Distance between pressure points in direction α
    • ",11)),t("li",null,[t("p",null,[s[51]||(s[51]=t("code",null,"A[α][β]",-1)),s[52]||(s[52]=e(": Interpolation weights from α-face centers ")),t("mjx-container",H,[(o(),n("svg",M,s[47]||(s[47]=[a('',1)]))),s[48]||(s[48]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"x"),t("mi",null,"I")])])],-1))]),s[53]||(s[53]=e(" to ")),t("mjx-container",L,[(o(),n("svg",C,s[49]||(s[49]=[a('',1)]))),s[50]||(s[50]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"x"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",null,"I"),t("mo",null,"±"),t("msub",null,[t("mi",null,"h"),t("mi",null,"β")])])])])],-1))])])])]),s[56]||(s[56]=t("p",null,"Note that the memory footprint of the redundant 1D-arrays above is negligible compared to the memory footprint of the 2D/3D-fields used in the code.",-1)),s[57]||(s[57]=t("p",null,[t("a",{href:"https://github.com/agdestein/IncompressibleNavierStokes.jl/blob/03acafa3566742b1f37d722d2207f5dc21d031c0/src/grid.jl#L79",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",N,[t("summary",null,[s[58]||(s[58]=t("a",{id:"IncompressibleNavierStokes.cosine_grid-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.cosine_grid-Tuple{Any, Any, Any}"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.cosine_grid")],-1)),s[59]||(s[59]=e()),l(i,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[62]||(s[62]=a('
    julia
    cosine_grid(a, b, N) -> Any
    Create a nonuniform grid of N + 1 points from a to b using a cosine profile, i.e.
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    julia
    max_size(grid) -> Any
    Get size of the largest grid element.
    source
    ',3))]),t("details",A,[t("summary",null,[s[68]||(s[68]=t("a",{id:"IncompressibleNavierStokes.stretched_grid",href:"#IncompressibleNavierStokes.stretched_grid"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.stretched_grid")],-1)),s[69]||(s[69]=e()),l(i,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[94]||(s[94]=a(`
    julia
    stretched_grid(a, b, N) -> Any
-stretched_grid(a, b, N, s) -> Any
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  • N[α]: Number of finite volumes in direction β, including ghost volumes
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    julia
    tanh_grid(a, b, N) -> Any
+tanh_grid(a, b, N, γ) -> Any
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    CPU(; static=false)
    Instantiate a CPU (multi-threaded) backend.
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 ;
     x,
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@@ -16,7 +16,7 @@ import{_ as r,c as n,j as t,a as e,G as l,a5 as a,B as d,o}from"./chunks/framewo
     workgroupsize,
     temperature,
     Re
-)
    Create problem setup (stored in a named tuple).
    source
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    temperature_equation(
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    Create problem setup (stored in a named tuple).
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    temperature_equation(
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@@ -25,7 +25,7 @@ import{_ as r,c as n,j as t,a as e,G as l,a5 as a,B as d,o}from"./chunks/framewo
     boundary_conditions,
     gdir,
     nondim_type
-)
    Create temperature equation setup (stored in a named tuple).
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-random_field(setup, t; A, kp, psolver, rng) -> Any
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    • A: Eddy amplitude scaling
    • kp: Peak energy wavenumber
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-velocityfield(setup, ufunc, t; psolver, doproject) -> Any
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    Create temperature equation setup (stored in a named tuple).
    The equation is parameterized by three dimensionless numbers (Prandtl number, Rayleigh number, and Gebhart number), and requires separate boundary conditions for the temperature field. The gdir keyword specifies the direction gravity, while non_dim_type specifies the type of non-dimensionalization.
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    random_field(setup; ...) -> Any
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    • A: Eddy amplitude scaling
    • kp: Peak energy wavenumber
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    temperaturefield(setup, tempfunc) -> Any
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    Create empty vector field.
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    velocityfield(setup, ufunc; ...) -> Any
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    Create divergence free velocity field u with boundary conditions at time t. The initial conditions of u[α] are specified by the function ufunc(α, x...).
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diff --git a/dev/assets/manual_solver.md.CSATxmgf.js b/dev/assets/manual_solver.md.CveGplQk.js
similarity index 84%
rename from dev/assets/manual_solver.md.CSATxmgf.js
rename to dev/assets/manual_solver.md.CveGplQk.js
index 884cf5a6d..faea5021b 100644
--- a/dev/assets/manual_solver.md.CSATxmgf.js
+++ b/dev/assets/manual_solver.md.CveGplQk.js
@@ -1,6 +1,6 @@
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@@ -16,7 +16,7 @@ import{_ as p,c as n,j as s,a as t,G as l,a5 as a,B as r,o}from"./chunks/framewo
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-)
    Solve unsteady problem using method.
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    as in San and Staples [12].
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@@ -40,7 +40,7 @@ import{_ as p,c as n,j as s,a as t,G as l,a5 as a,B as r,o}from"./chunks/framewo
 The summand is 3, the time is 1.2
 The summand is 4, the time is 1.6
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    Keyword arguments:
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    • nupdate: Show solution every nupdate time step.
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    ',5))]),s("details",x,[s("summary",null,[e[43]||(e[43]=s("a",{id:"IncompressibleNavierStokes.save_vtk-Tuple{Any}",href:"#IncompressibleNavierStokes.save_vtk-Tuple{Any}"},[s("span",{class:"jlbinding"},"IncompressibleNavierStokes.save_vtk")],-1)),e[44]||(e[44]=t()),l(i,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[45]||(e[45]=a('
    julia
    save_vtk(state; setup, filename, kwargs...)
    Save fields to vtk file.
    The kwargs are passed to snapshotsaver.
    source
    ',4))]),s("details",L,[s("summary",null,[e[46]||(e[46]=s("a",{id:"IncompressibleNavierStokes.snapshotsaver-Tuple{Any}",href:"#IncompressibleNavierStokes.snapshotsaver-Tuple{Any}"},[s("span",{class:"jlbinding"},"IncompressibleNavierStokes.snapshotsaver")],-1)),e[47]||(e[47]=t()),l(i,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[48]||(e[48]=a('
    julia
    snapshotsaver(state; setup, fieldnames, psolver)
    In the case of a 2D setup, the velocity field is saved as a 3D vector with a z-component of zero, as this seems to be preferred by ParaView.
    source
    ',3))]),s("details",w,[s("summary",null,[e[49]||(e[49]=s("a",{id:"IncompressibleNavierStokes.timelogger-Tuple{}",href:"#IncompressibleNavierStokes.timelogger-Tuple{}"},[s("span",{class:"jlbinding"},"IncompressibleNavierStokes.timelogger")],-1)),e[50]||(e[50]=t()),l(i,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[51]||(e[51]=a(`
    julia
    timelogger(
+The final sum (at time t=2.0) is 15
    source
    `,8))]),s("details",C,[s("summary",null,[e[40]||(e[40]=s("a",{id:"IncompressibleNavierStokes.realtimeplotter",href:"#IncompressibleNavierStokes.realtimeplotter"},[s("span",{class:"jlbinding"},"IncompressibleNavierStokes.realtimeplotter")],-1)),e[41]||(e[41]=a()),l(i,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),e[42]||(e[42]=t('
    Processor for plotting the solution in real time.
    Keyword arguments:
    • plot: Plot function.
    • nupdate: Show solution every nupdate time step.
    • displayfig: Display the figure at the start.
    • screen: If nothing, use default display. If GLMakie.screen() multiple plots can be displayed in separate windows like in MATLAB (see also GLMakie.closeall()).
    • displayupdates: Display the figure at every update (if using CairoMakie).
    • sleeptime: The sleeptime is slept at every update, to give Makie time to update the plot. Set this to nothing to skip sleeping.
    Additional kwargs are passed to the plot function.
    source
    ',5))]),s("details",x,[s("summary",null,[e[43]||(e[43]=s("a",{id:"IncompressibleNavierStokes.save_vtk-Tuple{Any}",href:"#IncompressibleNavierStokes.save_vtk-Tuple{Any}"},[s("span",{class:"jlbinding"},"IncompressibleNavierStokes.save_vtk")],-1)),e[44]||(e[44]=a()),l(i,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[45]||(e[45]=t('
    julia
    save_vtk(state; setup, filename, kwargs...)
    Save fields to vtk file.
    The kwargs are passed to snapshotsaver.
    source
    ',4))]),s("details",L,[s("summary",null,[e[46]||(e[46]=s("a",{id:"IncompressibleNavierStokes.snapshotsaver-Tuple{Any}",href:"#IncompressibleNavierStokes.snapshotsaver-Tuple{Any}"},[s("span",{class:"jlbinding"},"IncompressibleNavierStokes.snapshotsaver")],-1)),e[47]||(e[47]=a()),l(i,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[48]||(e[48]=t('
    julia
    snapshotsaver(state; setup, fieldnames, psolver)
    In the case of a 2D setup, the velocity field is saved as a 3D vector with a z-component of zero, as this seems to be preferred by ParaView.
    source
    ',3))]),s("details",w,[s("summary",null,[e[49]||(e[49]=s("a",{id:"IncompressibleNavierStokes.timelogger-Tuple{}",href:"#IncompressibleNavierStokes.timelogger-Tuple{}"},[s("span",{class:"jlbinding"},"IncompressibleNavierStokes.timelogger")],-1)),e[50]||(e[50]=a()),l(i,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[51]||(e[51]=t(`
    julia
    timelogger(
 ;
     showiter,
     showt,
@@ -48,4 +48,4 @@ import{_ as p,c as n,j as s,a as t,G as l,a5 as a,B as r,o}from"./chunks/framewo
     showmax,
     showspeed,
     nupdate
-) -> @NamedTuple{initialize::IncompressibleNavierStokes.var"#298#300"{Bool, Bool, Bool, Bool, Bool, Int64}, finalize::IncompressibleNavierStokes.var"#295#296"}
    Create processor that logs time step information.
    source
    `,3))]),s("details",S,[s("summary",null,[e[52]||(e[52]=s("a",{id:"IncompressibleNavierStokes.vtk_writer-Tuple{}",href:"#IncompressibleNavierStokes.vtk_writer-Tuple{}"},[s("span",{class:"jlbinding"},"IncompressibleNavierStokes.vtk_writer")],-1)),e[53]||(e[53]=t()),l(i,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[54]||(e[54]=a('
    julia
    vtk_writer(; setup, nupdate, dir, filename, kwargs...)
    Create processor that writes the solution every nupdate time steps to a VTK file. The resulting Paraview data collection file is stored in "$dir/$filename.pvd". The kwargs are passed to snapshotsaver.
    source
    ',3))])])}const Z=p(d,[["render",N]]);export{V as __pageData,Z as default};
+) -> @NamedTuple{initialize::IncompressibleNavierStokes.var"#298#300"{Bool, Bool, Bool, Bool, Bool, Int64}, finalize::IncompressibleNavierStokes.var"#295#296"}
    Create processor that logs time step information.
    source
    `,3))]),s("details",S,[s("summary",null,[e[52]||(e[52]=s("a",{id:"IncompressibleNavierStokes.vtk_writer-Tuple{}",href:"#IncompressibleNavierStokes.vtk_writer-Tuple{}"},[s("span",{class:"jlbinding"},"IncompressibleNavierStokes.vtk_writer")],-1)),e[53]||(e[53]=a()),l(i,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[54]||(e[54]=t('
    julia
    vtk_writer(; setup, nupdate, dir, filename, kwargs...)
    Create processor that writes the solution every nupdate time steps to a VTK file. The resulting Paraview data collection file is stored in "$dir/$filename.pvd". The kwargs are passed to snapshotsaver.
    source
    ',3))])])}const Z=p(d,[["render",N]]);export{V as __pageData,Z as default};
diff --git a/dev/assets/manual_solver.md.CSATxmgf.lean.js b/dev/assets/manual_solver.md.CveGplQk.lean.js
similarity index 84%
rename from dev/assets/manual_solver.md.CSATxmgf.lean.js
rename to dev/assets/manual_solver.md.CveGplQk.lean.js
index 884cf5a6d..faea5021b 100644
--- a/dev/assets/manual_solver.md.CSATxmgf.lean.js
+++ b/dev/assets/manual_solver.md.CveGplQk.lean.js
@@ -1,6 +1,6 @@
-import{_ as p,c as n,j as s,a as t,G as l,a5 as a,B as r,o}from"./chunks/framework.Dr1kDDE8.js";const V=JSON.parse('{"title":"Solvers","description":"","frontmatter":{},"headers":[],"relativePath":"manual/solver.md","filePath":"manual/solver.md","lastUpdated":null}'),d={name:"manual/solver.md"},h={class:"jldocstring custom-block",open:""},k={class:"jldocstring custom-block",open:""},c={class:"jldocstring custom-block",open:""},m={class:"jldocstring custom-block",open:""},g={class:"jldocstring custom-block",open:""},T={class:"jldocstring custom-block",open:""},Q={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},u={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.09ex"},xmlns:"http://www.w3.org/2000/svg",width:"5.703ex",height:"1.636ex",role:"img",focusable:"false",viewBox:"0 -683 2520.6 723","aria-hidden":"true"},y={class:"MathJax",jax:"SVG",display:"true",style:{direction:"ltr",display:"block","text-align":"center",margin:"1em 0",position:"relative"}},b={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-2.428ex"},xmlns:"http://www.w3.org/2000/svg",width:"26.783ex",height:"5.507ex",role:"img",focusable:"false",viewBox:"0 -1361 11837.9 2434.1","aria-hidden":"true"},E={class:"jldocstring custom-block",open:""},v={class:"jldocstring custom-block",open:""},f={class:"jldocstring custom-block",open:""},j={class:"jldocstring custom-block",open:""},F={class:"jldocstring custom-block",open:""},C={class:"jldocstring custom-block",open:""},x={class:"jldocstring custom-block",open:""},L={class:"jldocstring custom-block",open:""},w={class:"jldocstring custom-block",open:""},S={class:"jldocstring custom-block",open:""};function N(I,e,H,M,B,A){const i=r("Badge");return o(),n("div",null,[e[55]||(e[55]=s("h1",{id:"solvers",tabindex:"-1"},[t("Solvers "),s("a",{class:"header-anchor",href:"#solvers","aria-label":'Permalink to "Solvers"'},"​")],-1)),e[56]||(e[56]=s("h2",{id:"Solvers-2",tabindex:"-1"},[t("Solvers "),s("a",{class:"header-anchor",href:"#Solvers-2","aria-label":'Permalink to "Solvers {#Solvers-2}"'},"​")],-1)),s("details",h,[s("summary",null,[e[0]||(e[0]=s("a",{id:"IncompressibleNavierStokes.get_cfl_timestep!-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.get_cfl_timestep!-Tuple{Any, Any, Any}"},[s("span",{class:"jlbinding"},"IncompressibleNavierStokes.get_cfl_timestep!")],-1)),e[1]||(e[1]=t()),l(i,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[2]||(e[2]=a('
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    get_cfl_timestep!(buf, u, setup) -> Any
    Get proposed maximum time step for convection and diffusion terms.
    source
    ',3))]),s("details",k,[s("summary",null,[e[3]||(e[3]=s("a",{id:"IncompressibleNavierStokes.get_state-Tuple{Any}",href:"#IncompressibleNavierStokes.get_state-Tuple{Any}"},[s("span",{class:"jlbinding"},"IncompressibleNavierStokes.get_state")],-1)),e[4]||(e[4]=t()),l(i,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[5]||(e[5]=a(`
    julia
    get_state(
+import{_ as p,c as n,j as s,a,G as l,a5 as t,B as r,o}from"./chunks/framework.Dr1kDDE8.js";const V=JSON.parse('{"title":"Solvers","description":"","frontmatter":{},"headers":[],"relativePath":"manual/solver.md","filePath":"manual/solver.md","lastUpdated":null}'),d={name:"manual/solver.md"},h={class:"jldocstring custom-block",open:""},k={class:"jldocstring custom-block",open:""},c={class:"jldocstring custom-block",open:""},m={class:"jldocstring custom-block",open:""},g={class:"jldocstring custom-block",open:""},T={class:"jldocstring custom-block",open:""},Q={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},u={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.09ex"},xmlns:"http://www.w3.org/2000/svg",width:"5.703ex",height:"1.636ex",role:"img",focusable:"false",viewBox:"0 -683 2520.6 723","aria-hidden":"true"},y={class:"MathJax",jax:"SVG",display:"true",style:{direction:"ltr",display:"block","text-align":"center",margin:"1em 0",position:"relative"}},b={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-2.428ex"},xmlns:"http://www.w3.org/2000/svg",width:"26.783ex",height:"5.507ex",role:"img",focusable:"false",viewBox:"0 -1361 11837.9 2434.1","aria-hidden":"true"},E={class:"jldocstring custom-block",open:""},v={class:"jldocstring custom-block",open:""},f={class:"jldocstring custom-block",open:""},j={class:"jldocstring custom-block",open:""},F={class:"jldocstring custom-block",open:""},C={class:"jldocstring custom-block",open:""},x={class:"jldocstring custom-block",open:""},L={class:"jldocstring custom-block",open:""},w={class:"jldocstring custom-block",open:""},S={class:"jldocstring custom-block",open:""};function N(I,e,H,M,B,A){const i=r("Badge");return o(),n("div",null,[e[55]||(e[55]=s("h1",{id:"solvers",tabindex:"-1"},[a("Solvers "),s("a",{class:"header-anchor",href:"#solvers","aria-label":'Permalink to "Solvers"'},"​")],-1)),e[56]||(e[56]=s("h2",{id:"Solvers-2",tabindex:"-1"},[a("Solvers "),s("a",{class:"header-anchor",href:"#Solvers-2","aria-label":'Permalink to "Solvers {#Solvers-2}"'},"​")],-1)),s("details",h,[s("summary",null,[e[0]||(e[0]=s("a",{id:"IncompressibleNavierStokes.get_cfl_timestep!-Tuple{Any, Any, Any}",href:"#IncompressibleNavierStokes.get_cfl_timestep!-Tuple{Any, Any, Any}"},[s("span",{class:"jlbinding"},"IncompressibleNavierStokes.get_cfl_timestep!")],-1)),e[1]||(e[1]=a()),l(i,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[2]||(e[2]=t('
    julia
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    Get proposed maximum time step for convection and diffusion terms.
    source
    ',3))]),s("details",k,[s("summary",null,[e[3]||(e[3]=s("a",{id:"IncompressibleNavierStokes.get_state-Tuple{Any}",href:"#IncompressibleNavierStokes.get_state-Tuple{Any}"},[s("span",{class:"jlbinding"},"IncompressibleNavierStokes.get_state")],-1)),e[4]||(e[4]=a()),l(i,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[5]||(e[5]=t(`
    julia
    get_state(
     stepper
-) -> NamedTuple{(:u, :temp, :t, :n), <:NTuple{4, Any}}
    Get state (; u, temp, t, n) from stepper.
    source
    `,3))]),s("details",c,[s("summary",null,[e[6]||(e[6]=s("a",{id:"IncompressibleNavierStokes.solve_unsteady-Tuple{}",href:"#IncompressibleNavierStokes.solve_unsteady-Tuple{}"},[s("span",{class:"jlbinding"},"IncompressibleNavierStokes.solve_unsteady")],-1)),e[7]||(e[7]=t()),l(i,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[8]||(e[8]=a(`
    julia
    solve_unsteady(
+) -> NamedTuple{(:u, :temp, :t, :n), <:NTuple{4, Any}}
    Get state (; u, temp, t, n) from stepper.
    source
    `,3))]),s("details",c,[s("summary",null,[e[6]||(e[6]=s("a",{id:"IncompressibleNavierStokes.solve_unsteady-Tuple{}",href:"#IncompressibleNavierStokes.solve_unsteady-Tuple{}"},[s("span",{class:"jlbinding"},"IncompressibleNavierStokes.solve_unsteady")],-1)),e[7]||(e[7]=a()),l(i,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[8]||(e[8]=t(`
    julia
    solve_unsteady(
 ;
     setup,
     tlims,
@@ -16,7 +16,7 @@ import{_ as p,c as n,j as s,a as t,G as l,a5 as a,B as r,o}from"./chunks/framewo
     processors,
     θ,
     cache
-)
    Solve unsteady problem using method.
    If Δt is a real number, it is rounded such that (t_end - t_start) / Δt is an integer. If Δt = nothing, the time step is chosen every n_adapt_Δt iteration with CFL-number cfl. If Δt_min is given, the adaptive time step never goes below it.
    The processors are called after every time step.
    Note that the state observable passed to the processor.initialize function contains vector living on the device, and you may have to move them back to the host using Array(u) in the processor.
    Return (; u, t), outputs, where outputs is a named tuple with the outputs of processors with the same field names.
    source
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    Animate a plot of the solution every update iteration. The animation is saved to path, which should have one of the following extensions:
    • ".mkv"
    • ".mp4"
    • ".webm"
    • ".gif"
    The plot is determined by a plotter processor. Additional kwargs are passed to plot.
    source
    ',4))]),s("details",g,[s("summary",null,[e[12]||(e[12]=s("a",{id:"IncompressibleNavierStokes.energy_history_plot",href:"#IncompressibleNavierStokes.energy_history_plot"},[s("span",{class:"jlbinding"},"IncompressibleNavierStokes.energy_history_plot")],-1)),e[13]||(e[13]=t()),l(i,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),e[14]||(e[14]=s("p",null,"Create energy history plot.",-1)),e[15]||(e[15]=s("p",null,[s("a",{href:"https://github.com/agdestein/IncompressibleNavierStokes.jl/blob/03acafa3566742b1f37d722d2207f5dc21d031c0/src/processors.jl#L392",target:"_blank",rel:"noreferrer"},"source")],-1))]),s("details",T,[s("summary",null,[e[16]||(e[16]=s("a",{id:"IncompressibleNavierStokes.energy_spectrum_plot",href:"#IncompressibleNavierStokes.energy_spectrum_plot"},[s("span",{class:"jlbinding"},"IncompressibleNavierStokes.energy_spectrum_plot")],-1)),e[17]||(e[17]=t()),l(i,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s("p",null,[e[20]||(e[20]=t("Create energy spectrum plot. The energy at a scalar wavenumber level ")),s("mjx-container",Q,[(o(),n("svg",u,e[18]||(e[18]=[a('',1)]))),e[19]||(e[19]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"κ"),s("mo",null,"∈"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",{mathvariant:"double-struck"},"N")])])],-1))]),e[21]||(e[21]=t(" is defined by"))]),s("mjx-container",y,[(o(),n("svg",b,e[22]||(e[22]=[a('',1)]))),e[23]||(e[23]=s("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("mrow",{"data-mjx-texclass":"ORD"},[s("mover",null,[s("mi",null,"e"),s("mo",{stretchy:"false"},"^")])]),s("mo",{stretchy:"false"},"("),s("mi",null,"κ"),s("mo",{stretchy:"false"},")"),s("mo",null,"="),s("msub",null,[s("mo",{"data-mjx-texclass":"OP"},"∫"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",null,"κ"),s("mo",null,"≤"),s("mo",{"data-mjx-texclass":"ORD"},"∥"),s("mi",null,"k"),s("msub",null,[s("mo",{"data-mjx-texclass":"ORD"},"∥"),s("mn",null,"2")]),s("mo",null,"<"),s("mi",null,"κ"),s("mo",null,"+"),s("mn",null,"1")])]),s("mo",{"data-mjx-texclass":"ORD",stretchy:"false"},"|"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mover",null,[s("mi",null,"e"),s("mo",{stretchy:"false"},"^")])]),s("mo",{stretchy:"false"},"("),s("mi",null,"k"),s("mo",{stretchy:"false"},")"),s("mo",{"data-mjx-texclass":"ORD",stretchy:"false"},"|"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",{mathvariant:"normal"},"d")]),s("mi",null,"k"),s("mo",null,",")])],-1))]),e[24]||(e[24]=a('
    as in San and Staples [12].
    Keyword arguments:
    • sloperange = [0.6, 0.9]: Percentage (between 0 and 1) of x-axis where the slope is plotted.
    • slopeoffset = 1.3: How far above the energy spectrum the inertial slope is plotted.
    • kwargs...: They are passed to observespectrum.
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    Plot state field in pressure points. If state is Observable, then the plot is interactive.
    Available fieldnames are:
    • :velocity,
    • :vorticity,
    • :streamfunction,
    • :pressure.
    Available plot types for 2D are:
    • heatmap (default),
    • image,
    • contour,
    • contourf.
    Available plot types for 3D are:
    • contour (default).
    The alpha value gets passed to contour in 3D.
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    julia
    fieldsaver(; setup, nupdate)
    Create processor that stores the solution and time every nupdate time step.
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    Observe field fieldname at pressure points.
    source
    ',3))]),s("details",j,[s("summary",null,[e[34]||(e[34]=s("a",{id:"IncompressibleNavierStokes.observespectrum-Tuple{Any}",href:"#IncompressibleNavierStokes.observespectrum-Tuple{Any}"},[s("span",{class:"jlbinding"},"IncompressibleNavierStokes.observespectrum")],-1)),e[35]||(e[35]=t()),l(i,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[36]||(e[36]=a('
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    Observe energy spectrum of state.
    source
    ',3))]),s("details",F,[s("summary",null,[e[37]||(e[37]=s("a",{id:"IncompressibleNavierStokes.processor",href:"#IncompressibleNavierStokes.processor"},[s("span",{class:"jlbinding"},"IncompressibleNavierStokes.processor")],-1)),e[38]||(e[38]=t()),l(i,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),e[39]||(e[39]=a(`
    julia
    processor(
+)
    Solve unsteady problem using method.
    If Δt is a real number, it is rounded such that (t_end - t_start) / Δt is an integer. If Δt = nothing, the time step is chosen every n_adapt_Δt iteration with CFL-number cfl. If Δt_min is given, the adaptive time step never goes below it.
    The processors are called after every time step.
    Note that the state observable passed to the processor.initialize function contains vector living on the device, and you may have to move them back to the host using Array(u) in the processor.
    Return (; u, t), outputs, where outputs is a named tuple with the outputs of processors with the same field names.
    source
    `,7))]),e[57]||(e[57]=s("h2",{id:"processors",tabindex:"-1"},[a("Processors "),s("a",{class:"header-anchor",href:"#processors","aria-label":'Permalink to "Processors"'},"​")],-1)),e[58]||(e[58]=s("p",null,[a("Processors can be used to process the solution in "),s("a",{href:"/IncompressibleNavierStokes.jl/dev/manual/solver#IncompressibleNavierStokes.solve_unsteady-Tuple{}"},[s("code",null,"solve_unsteady")]),a(" after every time step.")],-1)),s("details",m,[s("summary",null,[e[9]||(e[9]=s("a",{id:"IncompressibleNavierStokes.animator",href:"#IncompressibleNavierStokes.animator"},[s("span",{class:"jlbinding"},"IncompressibleNavierStokes.animator")],-1)),e[10]||(e[10]=a()),l(i,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),e[11]||(e[11]=t('
    Animate a plot of the solution every update iteration. The animation is saved to path, which should have one of the following extensions:
    • ".mkv"
    • ".mp4"
    • ".webm"
    • ".gif"
    The plot is determined by a plotter processor. Additional kwargs are passed to plot.
    source
    ',4))]),s("details",g,[s("summary",null,[e[12]||(e[12]=s("a",{id:"IncompressibleNavierStokes.energy_history_plot",href:"#IncompressibleNavierStokes.energy_history_plot"},[s("span",{class:"jlbinding"},"IncompressibleNavierStokes.energy_history_plot")],-1)),e[13]||(e[13]=a()),l(i,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),e[14]||(e[14]=s("p",null,"Create energy history plot.",-1)),e[15]||(e[15]=s("p",null,[s("a",{href:"https://github.com/agdestein/IncompressibleNavierStokes.jl/blob/a63d20e3a410f14ba6cc612a27e89f8974019d9f/src/processors.jl#L392",target:"_blank",rel:"noreferrer"},"source")],-1))]),s("details",T,[s("summary",null,[e[16]||(e[16]=s("a",{id:"IncompressibleNavierStokes.energy_spectrum_plot",href:"#IncompressibleNavierStokes.energy_spectrum_plot"},[s("span",{class:"jlbinding"},"IncompressibleNavierStokes.energy_spectrum_plot")],-1)),e[17]||(e[17]=a()),l(i,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s("p",null,[e[20]||(e[20]=a("Create energy spectrum plot. The energy at a scalar wavenumber level ")),s("mjx-container",Q,[(o(),n("svg",u,e[18]||(e[18]=[t('',1)]))),e[19]||(e[19]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"κ"),s("mo",null,"∈"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",{mathvariant:"double-struck"},"N")])])],-1))]),e[21]||(e[21]=a(" is defined by"))]),s("mjx-container",y,[(o(),n("svg",b,e[22]||(e[22]=[t('',1)]))),e[23]||(e[23]=s("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("mrow",{"data-mjx-texclass":"ORD"},[s("mover",null,[s("mi",null,"e"),s("mo",{stretchy:"false"},"^")])]),s("mo",{stretchy:"false"},"("),s("mi",null,"κ"),s("mo",{stretchy:"false"},")"),s("mo",null,"="),s("msub",null,[s("mo",{"data-mjx-texclass":"OP"},"∫"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",null,"κ"),s("mo",null,"≤"),s("mo",{"data-mjx-texclass":"ORD"},"∥"),s("mi",null,"k"),s("msub",null,[s("mo",{"data-mjx-texclass":"ORD"},"∥"),s("mn",null,"2")]),s("mo",null,"<"),s("mi",null,"κ"),s("mo",null,"+"),s("mn",null,"1")])]),s("mo",{"data-mjx-texclass":"ORD",stretchy:"false"},"|"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mover",null,[s("mi",null,"e"),s("mo",{stretchy:"false"},"^")])]),s("mo",{stretchy:"false"},"("),s("mi",null,"k"),s("mo",{stretchy:"false"},")"),s("mo",{"data-mjx-texclass":"ORD",stretchy:"false"},"|"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",{mathvariant:"normal"},"d")]),s("mi",null,"k"),s("mo",null,",")])],-1))]),e[24]||(e[24]=t('
    as in San and Staples [12].
    Keyword arguments:
    • sloperange = [0.6, 0.9]: Percentage (between 0 and 1) of x-axis where the slope is plotted.
    • slopeoffset = 1.3: How far above the energy spectrum the inertial slope is plotted.
    • kwargs...: They are passed to observespectrum.
    source
    ',4))]),s("details",E,[s("summary",null,[e[25]||(e[25]=s("a",{id:"IncompressibleNavierStokes.fieldplot",href:"#IncompressibleNavierStokes.fieldplot"},[s("span",{class:"jlbinding"},"IncompressibleNavierStokes.fieldplot")],-1)),e[26]||(e[26]=a()),l(i,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),e[27]||(e[27]=t('
    Plot state field in pressure points. If state is Observable, then the plot is interactive.
    Available fieldnames are:
    • :velocity,
    • :vorticity,
    • :streamfunction,
    • :pressure.
    Available plot types for 2D are:
    • heatmap (default),
    • image,
    • contour,
    • contourf.
    Available plot types for 3D are:
    • contour (default).
    The alpha value gets passed to contour in 3D.
    source
    ',9))]),s("details",v,[s("summary",null,[e[28]||(e[28]=s("a",{id:"IncompressibleNavierStokes.fieldsaver-Tuple{}",href:"#IncompressibleNavierStokes.fieldsaver-Tuple{}"},[s("span",{class:"jlbinding"},"IncompressibleNavierStokes.fieldsaver")],-1)),e[29]||(e[29]=a()),l(i,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[30]||(e[30]=t('
    julia
    fieldsaver(; setup, nupdate)
    Create processor that stores the solution and time every nupdate time step.
    source
    ',3))]),s("details",f,[s("summary",null,[e[31]||(e[31]=s("a",{id:"IncompressibleNavierStokes.observefield-Tuple{Any}",href:"#IncompressibleNavierStokes.observefield-Tuple{Any}"},[s("span",{class:"jlbinding"},"IncompressibleNavierStokes.observefield")],-1)),e[32]||(e[32]=a()),l(i,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[33]||(e[33]=t('
    julia
    observefield(state; setup, fieldname, logtol, psolver)
    Observe field fieldname at pressure points.
    source
    ',3))]),s("details",j,[s("summary",null,[e[34]||(e[34]=s("a",{id:"IncompressibleNavierStokes.observespectrum-Tuple{Any}",href:"#IncompressibleNavierStokes.observespectrum-Tuple{Any}"},[s("span",{class:"jlbinding"},"IncompressibleNavierStokes.observespectrum")],-1)),e[35]||(e[35]=a()),l(i,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[36]||(e[36]=t('
    julia
    observespectrum(state; setup, npoint, a)
    Observe energy spectrum of state.
    source
    ',3))]),s("details",F,[s("summary",null,[e[37]||(e[37]=s("a",{id:"IncompressibleNavierStokes.processor",href:"#IncompressibleNavierStokes.processor"},[s("span",{class:"jlbinding"},"IncompressibleNavierStokes.processor")],-1)),e[38]||(e[38]=a()),l(i,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),e[39]||(e[39]=t(`
    julia
    processor(
     initialize
 ) -> NamedTuple{(:initialize, :finalize), <:Tuple{Any, IncompressibleNavierStokes.var"#295#296"}}
 processor(
@@ -40,7 +40,7 @@ import{_ as p,c as n,j as s,a as t,G as l,a5 as a,B as r,o}from"./chunks/framewo
 The summand is 3, the time is 1.2
 The summand is 4, the time is 1.6
 The summand is 5, the time is 2.0
-The final sum (at time t=2.0) is 15
    source
    `,8))]),s("details",C,[s("summary",null,[e[40]||(e[40]=s("a",{id:"IncompressibleNavierStokes.realtimeplotter",href:"#IncompressibleNavierStokes.realtimeplotter"},[s("span",{class:"jlbinding"},"IncompressibleNavierStokes.realtimeplotter")],-1)),e[41]||(e[41]=t()),l(i,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),e[42]||(e[42]=a('
    Processor for plotting the solution in real time.
    Keyword arguments:
    • plot: Plot function.
    • nupdate: Show solution every nupdate time step.
    • displayfig: Display the figure at the start.
    • screen: If nothing, use default display. If GLMakie.screen() multiple plots can be displayed in separate windows like in MATLAB (see also GLMakie.closeall()).
    • displayupdates: Display the figure at every update (if using CairoMakie).
    • sleeptime: The sleeptime is slept at every update, to give Makie time to update the plot. Set this to nothing to skip sleeping.
    Additional kwargs are passed to the plot function.
    source
    ',5))]),s("details",x,[s("summary",null,[e[43]||(e[43]=s("a",{id:"IncompressibleNavierStokes.save_vtk-Tuple{Any}",href:"#IncompressibleNavierStokes.save_vtk-Tuple{Any}"},[s("span",{class:"jlbinding"},"IncompressibleNavierStokes.save_vtk")],-1)),e[44]||(e[44]=t()),l(i,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[45]||(e[45]=a('
    julia
    save_vtk(state; setup, filename, kwargs...)
    Save fields to vtk file.
    The kwargs are passed to snapshotsaver.
    source
    ',4))]),s("details",L,[s("summary",null,[e[46]||(e[46]=s("a",{id:"IncompressibleNavierStokes.snapshotsaver-Tuple{Any}",href:"#IncompressibleNavierStokes.snapshotsaver-Tuple{Any}"},[s("span",{class:"jlbinding"},"IncompressibleNavierStokes.snapshotsaver")],-1)),e[47]||(e[47]=t()),l(i,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[48]||(e[48]=a('
    julia
    snapshotsaver(state; setup, fieldnames, psolver)
    In the case of a 2D setup, the velocity field is saved as a 3D vector with a z-component of zero, as this seems to be preferred by ParaView.
    source
    ',3))]),s("details",w,[s("summary",null,[e[49]||(e[49]=s("a",{id:"IncompressibleNavierStokes.timelogger-Tuple{}",href:"#IncompressibleNavierStokes.timelogger-Tuple{}"},[s("span",{class:"jlbinding"},"IncompressibleNavierStokes.timelogger")],-1)),e[50]||(e[50]=t()),l(i,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[51]||(e[51]=a(`
    julia
    timelogger(
+The final sum (at time t=2.0) is 15
    source
    `,8))]),s("details",C,[s("summary",null,[e[40]||(e[40]=s("a",{id:"IncompressibleNavierStokes.realtimeplotter",href:"#IncompressibleNavierStokes.realtimeplotter"},[s("span",{class:"jlbinding"},"IncompressibleNavierStokes.realtimeplotter")],-1)),e[41]||(e[41]=a()),l(i,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),e[42]||(e[42]=t('
    Processor for plotting the solution in real time.
    Keyword arguments:
    • plot: Plot function.
    • nupdate: Show solution every nupdate time step.
    • displayfig: Display the figure at the start.
    • screen: If nothing, use default display. If GLMakie.screen() multiple plots can be displayed in separate windows like in MATLAB (see also GLMakie.closeall()).
    • displayupdates: Display the figure at every update (if using CairoMakie).
    • sleeptime: The sleeptime is slept at every update, to give Makie time to update the plot. Set this to nothing to skip sleeping.
    Additional kwargs are passed to the plot function.
    source
    ',5))]),s("details",x,[s("summary",null,[e[43]||(e[43]=s("a",{id:"IncompressibleNavierStokes.save_vtk-Tuple{Any}",href:"#IncompressibleNavierStokes.save_vtk-Tuple{Any}"},[s("span",{class:"jlbinding"},"IncompressibleNavierStokes.save_vtk")],-1)),e[44]||(e[44]=a()),l(i,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[45]||(e[45]=t('
    julia
    save_vtk(state; setup, filename, kwargs...)
    Save fields to vtk file.
    The kwargs are passed to snapshotsaver.
    source
    ',4))]),s("details",L,[s("summary",null,[e[46]||(e[46]=s("a",{id:"IncompressibleNavierStokes.snapshotsaver-Tuple{Any}",href:"#IncompressibleNavierStokes.snapshotsaver-Tuple{Any}"},[s("span",{class:"jlbinding"},"IncompressibleNavierStokes.snapshotsaver")],-1)),e[47]||(e[47]=a()),l(i,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[48]||(e[48]=t('
    julia
    snapshotsaver(state; setup, fieldnames, psolver)
    In the case of a 2D setup, the velocity field is saved as a 3D vector with a z-component of zero, as this seems to be preferred by ParaView.
    source
    ',3))]),s("details",w,[s("summary",null,[e[49]||(e[49]=s("a",{id:"IncompressibleNavierStokes.timelogger-Tuple{}",href:"#IncompressibleNavierStokes.timelogger-Tuple{}"},[s("span",{class:"jlbinding"},"IncompressibleNavierStokes.timelogger")],-1)),e[50]||(e[50]=a()),l(i,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[51]||(e[51]=t(`
    julia
    timelogger(
 ;
     showiter,
     showt,
@@ -48,4 +48,4 @@ import{_ as p,c as n,j as s,a as t,G as l,a5 as a,B as r,o}from"./chunks/framewo
     showmax,
     showspeed,
     nupdate
-) -> @NamedTuple{initialize::IncompressibleNavierStokes.var"#298#300"{Bool, Bool, Bool, Bool, Bool, Int64}, finalize::IncompressibleNavierStokes.var"#295#296"}
    Create processor that logs time step information.
    source
    `,3))]),s("details",S,[s("summary",null,[e[52]||(e[52]=s("a",{id:"IncompressibleNavierStokes.vtk_writer-Tuple{}",href:"#IncompressibleNavierStokes.vtk_writer-Tuple{}"},[s("span",{class:"jlbinding"},"IncompressibleNavierStokes.vtk_writer")],-1)),e[53]||(e[53]=t()),l(i,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[54]||(e[54]=a('
    julia
    vtk_writer(; setup, nupdate, dir, filename, kwargs...)
    Create processor that writes the solution every nupdate time steps to a VTK file. The resulting Paraview data collection file is stored in "$dir/$filename.pvd". The kwargs are passed to snapshotsaver.
    source
    ',3))])])}const Z=p(d,[["render",N]]);export{V as __pageData,Z as default};
+) -> @NamedTuple{initialize::IncompressibleNavierStokes.var"#298#300"{Bool, Bool, Bool, Bool, Bool, Int64}, finalize::IncompressibleNavierStokes.var"#295#296"}
    Create processor that logs time step information.
    source
    `,3))]),s("details",S,[s("summary",null,[e[52]||(e[52]=s("a",{id:"IncompressibleNavierStokes.vtk_writer-Tuple{}",href:"#IncompressibleNavierStokes.vtk_writer-Tuple{}"},[s("span",{class:"jlbinding"},"IncompressibleNavierStokes.vtk_writer")],-1)),e[53]||(e[53]=a()),l(i,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[54]||(e[54]=t('
    julia
    vtk_writer(; setup, nupdate, dir, filename, kwargs...)
    Create processor that writes the solution every nupdate time steps to a VTK file. The resulting Paraview data collection file is stored in "$dir/$filename.pvd". The kwargs are passed to snapshotsaver.
    source
    ',3))])])}const Z=p(d,[["render",N]]);export{V as __pageData,Z as default};
diff --git a/dev/assets/manual_temperature.md.CvylYS1B.js b/dev/assets/manual_temperature.md.DGNhfuFk.js
similarity index 93%
rename from dev/assets/manual_temperature.md.CvylYS1B.js
rename to dev/assets/manual_temperature.md.DGNhfuFk.js
index ee2cfd6ac..ab5aadaac 100644
--- a/dev/assets/manual_temperature.md.CvylYS1B.js
+++ b/dev/assets/manual_temperature.md.DGNhfuFk.js
@@ -1,7 +1,7 @@
-import{_ as t,c as n,a5 as a,j as s,a as p,G as r,B as l,o}from"./chunks/framework.Dr1kDDE8.js";const b=JSON.parse('{"title":"Temperature equation","description":"","frontmatter":{},"headers":[],"relativePath":"manual/temperature.md","filePath":"manual/temperature.md","lastUpdated":null}'),d={name:"manual/temperature.md"},h={class:"jldocstring custom-block",open:""};function k(u,e,c,E,m,g){const i=l("Badge");return o(),n("div",null,[e[3]||(e[3]=a(`
    Temperature equation ​
    IncompressibleNavierStokes.jl supports adding a temperature equation, which is coupled back to the momentum equation through a gravity term [8].
    To enable the temperature equation, you need to set the temperature keyword in setup:
    julia
    setup = Setup(;
+import{_ as t,c as n,a5 as a,j as s,a as p,G as r,B as l,o}from"./chunks/framework.Dr1kDDE8.js";const b=JSON.parse('{"title":"Temperature equation","description":"","frontmatter":{},"headers":[],"relativePath":"manual/temperature.md","filePath":"manual/temperature.md","lastUpdated":null}'),h={name:"manual/temperature.md"},d={class:"jldocstring custom-block",open:""};function k(u,e,c,E,m,g){const i=l("Badge");return o(),n("div",null,[e[3]||(e[3]=a(`
    Temperature equation ​
    IncompressibleNavierStokes.jl supports adding a temperature equation, which is coupled back to the momentum equation through a gravity term [8].
    To enable the temperature equation, you need to set the temperature keyword in setup:
    julia
    setup = Setup(;
     kwargs...,
     temperature = temperature_equation(; kwargs...),
-)
    where temperature_equation can be configured as follows:
    `,5)),s("details",h,[s("summary",null,[e[0]||(e[0]=s("a",{id:"IncompressibleNavierStokes.temperature_equation-manual-temperature",href:"#IncompressibleNavierStokes.temperature_equation-manual-temperature"},[s("span",{class:"jlbinding"},"IncompressibleNavierStokes.temperature_equation")],-1)),e[1]||(e[1]=p()),r(i,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),e[2]||(e[2]=a(`
    julia
    temperature_equation(
+)
    where temperature_equation can be configured as follows:
    `,5)),s("details",d,[s("summary",null,[e[0]||(e[0]=s("a",{id:"IncompressibleNavierStokes.temperature_equation-manual-temperature",href:"#IncompressibleNavierStokes.temperature_equation-manual-temperature"},[s("span",{class:"jlbinding"},"IncompressibleNavierStokes.temperature_equation")],-1)),e[1]||(e[1]=p()),r(i,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),e[2]||(e[2]=a(`
    julia
    temperature_equation(
 ;
     Pr,
     Ra,
@@ -10,4 +10,4 @@ import{_ as t,c as n,a5 as a,j as s,a as p,G as r,B as l,o}from"./chunks/framewo
     boundary_conditions,
     gdir,
     nondim_type
-)
    Create temperature equation setup (stored in a named tuple).
    The equation is parameterized by three dimensionless numbers (Prandtl number, Rayleigh number, and Gebhart number), and requires separate boundary conditions for the temperature field. The gdir keyword specifies the direction gravity, while non_dim_type specifies the type of non-dimensionalization.
    source
    `,4))])])}const f=t(d,[["render",k]]);export{b as __pageData,f as default};
+)
    Create temperature equation setup (stored in a named tuple).
    The equation is parameterized by three dimensionless numbers (Prandtl number, Rayleigh number, and Gebhart number), and requires separate boundary conditions for the temperature field. The gdir keyword specifies the direction gravity, while non_dim_type specifies the type of non-dimensionalization.
    source
    `,4))])])}const f=t(h,[["render",k]]);export{b as __pageData,f as default};
diff --git a/dev/assets/manual_temperature.md.CvylYS1B.lean.js b/dev/assets/manual_temperature.md.DGNhfuFk.lean.js
similarity index 93%
rename from dev/assets/manual_temperature.md.CvylYS1B.lean.js
rename to dev/assets/manual_temperature.md.DGNhfuFk.lean.js
index ee2cfd6ac..ab5aadaac 100644
--- a/dev/assets/manual_temperature.md.CvylYS1B.lean.js
+++ b/dev/assets/manual_temperature.md.DGNhfuFk.lean.js
@@ -1,7 +1,7 @@
-import{_ as t,c as n,a5 as a,j as s,a as p,G as r,B as l,o}from"./chunks/framework.Dr1kDDE8.js";const b=JSON.parse('{"title":"Temperature equation","description":"","frontmatter":{},"headers":[],"relativePath":"manual/temperature.md","filePath":"manual/temperature.md","lastUpdated":null}'),d={name:"manual/temperature.md"},h={class:"jldocstring custom-block",open:""};function k(u,e,c,E,m,g){const i=l("Badge");return o(),n("div",null,[e[3]||(e[3]=a(`
    Temperature equation ​
    IncompressibleNavierStokes.jl supports adding a temperature equation, which is coupled back to the momentum equation through a gravity term [8].
    To enable the temperature equation, you need to set the temperature keyword in setup:
    julia
    setup = Setup(;
+import{_ as t,c as n,a5 as a,j as s,a as p,G as r,B as l,o}from"./chunks/framework.Dr1kDDE8.js";const b=JSON.parse('{"title":"Temperature equation","description":"","frontmatter":{},"headers":[],"relativePath":"manual/temperature.md","filePath":"manual/temperature.md","lastUpdated":null}'),h={name:"manual/temperature.md"},d={class:"jldocstring custom-block",open:""};function k(u,e,c,E,m,g){const i=l("Badge");return o(),n("div",null,[e[3]||(e[3]=a(`
    Temperature equation ​
    IncompressibleNavierStokes.jl supports adding a temperature equation, which is coupled back to the momentum equation through a gravity term [8].
    To enable the temperature equation, you need to set the temperature keyword in setup:
    julia
    setup = Setup(;
     kwargs...,
     temperature = temperature_equation(; kwargs...),
-)
    where temperature_equation can be configured as follows:
    `,5)),s("details",h,[s("summary",null,[e[0]||(e[0]=s("a",{id:"IncompressibleNavierStokes.temperature_equation-manual-temperature",href:"#IncompressibleNavierStokes.temperature_equation-manual-temperature"},[s("span",{class:"jlbinding"},"IncompressibleNavierStokes.temperature_equation")],-1)),e[1]||(e[1]=p()),r(i,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),e[2]||(e[2]=a(`
    julia
    temperature_equation(
+)
    where temperature_equation can be configured as follows:
    `,5)),s("details",d,[s("summary",null,[e[0]||(e[0]=s("a",{id:"IncompressibleNavierStokes.temperature_equation-manual-temperature",href:"#IncompressibleNavierStokes.temperature_equation-manual-temperature"},[s("span",{class:"jlbinding"},"IncompressibleNavierStokes.temperature_equation")],-1)),e[1]||(e[1]=p()),r(i,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),e[2]||(e[2]=a(`
    julia
    temperature_equation(
 ;
     Pr,
     Ra,
@@ -10,4 +10,4 @@ import{_ as t,c as n,a5 as a,j as s,a as p,G as r,B as l,o}from"./chunks/framewo
     boundary_conditions,
     gdir,
     nondim_type
-)
    Create temperature equation setup (stored in a named tuple).
    The equation is parameterized by three dimensionless numbers (Prandtl number, Rayleigh number, and Gebhart number), and requires separate boundary conditions for the temperature field. The gdir keyword specifies the direction gravity, while non_dim_type specifies the type of non-dimensionalization.
    source
    `,4))])])}const f=t(d,[["render",k]]);export{b as __pageData,f as default};
+)
    Create temperature equation setup (stored in a named tuple).
    The equation is parameterized by three dimensionless numbers (Prandtl number, Rayleigh number, and Gebhart number), and requires separate boundary conditions for the temperature field. The gdir keyword specifies the direction gravity, while non_dim_type specifies the type of non-dimensionalization.
    source
    `,4))])])}const f=t(h,[["render",k]]);export{b as __pageData,f as default};
diff --git a/dev/assets/manual_time.md.3eTgRN26.js b/dev/assets/manual_time.md.BJBCC_BN.js
similarity index 99%
rename from dev/assets/manual_time.md.3eTgRN26.js
rename to dev/assets/manual_time.md.BJBCC_BN.js
index d664e51e2..85c06a11c 100644
--- a/dev/assets/manual_time.md.3eTgRN26.js
+++ b/dev/assets/manual_time.md.BJBCC_BN.js
@@ -1,174 +1,174 @@
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We define ")),t("mjx-container",F,[(l(),Q("svg",R,a[27]||(a[27]=[e('',1)]))),a[28]||(a[28]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msup",null,[t("mi",null,"u"),t("mi",null,"n")]),t("mo",null,"≈"),t("mi",null,"u"),t("mo",{stretchy:"false"},"("),t("msup",null,[t("mi",null,"t"),t("mi",null,"n")]),t("mo",{stretchy:"false"},")")])],-1))]),a[50]||(a[50]=s(" as an approximation to the exact discrete velocity field ")),t("mjx-container",C,[(l(),Q("svg",N,a[29]||(a[29]=[e('',1)]))),a[30]||(a[30]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"u"),t("mo",{stretchy:"false"},"("),t("msup",null,[t("mi",null,"t"),t("mi",null,"n")]),t("mo",{stretchy:"false"},")")])],-1))]),a[51]||(a[51]=s(", with ")),t("mjx-container",K,[(l(),Q("svg",A,a[31]||(a[31]=[e('',1)]))),a[32]||(a[32]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msup",null,[t("mi",null,"u"),t("mn",null,"0")]),t("mo",null,"="),t("mi",null,"u"),t("mo",{stretchy:"false"},"("),t("mn",null,"0"),t("mo",{stretchy:"false"},")")])],-1))]),a[52]||(a[52]=s(" starting from the exact initial conditions. We say that the time integration scheme (definition of ")),t("mjx-container",B,[(l(),Q("svg",O,a[33]||(a[33]=[e('',1)]))),a[34]||(a[34]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msup",null,[t("mi",null,"u"),t("mi",null,"n")])])],-1))]),a[53]||(a[53]=s(") is accurate to the order ")),t("mjx-container",P,[(l(),Q("svg",G,a[35]||(a[35]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D45F",d:"M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 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")),t("mjx-container",D,[(l(),Q("svg",S,a[23]||(a[23]=[e('',1)]))),a[24]||(a[24]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msup",null,[t("mi",null,"t"),t("mi",null,"N")]),t("mo",null,"="),t("mi",null,"T")])],-1))]),a[48]||(a[48]=s(", and increment ")),t("mjx-container",V,[(l(),Q("svg",I,a[25]||(a[25]=[e('',1)]))),a[26]||(a[26]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 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We define ")),t("mjx-container",F,[(l(),Q("svg",R,a[27]||(a[27]=[e('',1)]))),a[28]||(a[28]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msup",null,[t("mi",null,"u"),t("mi",null,"n")]),t("mo",null,"≈"),t("mi",null,"u"),t("mo",{stretchy:"false"},"("),t("msup",null,[t("mi",null,"t"),t("mi",null,"n")]),t("mo",{stretchy:"false"},")")])],-1))]),a[50]||(a[50]=s(" as an approximation to the exact discrete velocity field ")),t("mjx-container",C,[(l(),Q("svg",N,a[29]||(a[29]=[e('',1)]))),a[30]||(a[30]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"u"),t("mo",{stretchy:"false"},"("),t("msup",null,[t("mi",null,"t"),t("mi",null,"n")]),t("mo",{stretchy:"false"},")")])],-1))]),a[51]||(a[51]=s(", with ")),t("mjx-container",K,[(l(),Q("svg",A,a[31]||(a[31]=[e('',1)]))),a[32]||(a[32]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msup",null,[t("mi",null,"u"),t("mn",null,"0")]),t("mo",null,"="),t("mi",null,"u"),t("mo",{stretchy:"false"},"("),t("mn",null,"0"),t("mo",{stretchy:"false"},")")])],-1))]),a[52]||(a[52]=s(" starting from the exact initial conditions. We say that the time integration scheme (definition of ")),t("mjx-container",B,[(l(),Q("svg",O,a[33]||(a[33]=[e('',1)]))),a[34]||(a[34]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msup",null,[t("mi",null,"u"),t("mi",null,"n")])])],-1))]),a[53]||(a[53]=s(") is accurate to the order ")),t("mjx-container",P,[(l(),Q("svg",G,a[35]||(a[35]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D45F",d:"M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 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jlType",text:"Type"})]),a[59]||(a[59]=e('
    julia
    abstract type AbstractODEMethod{T}
    Abstract ODE method.
    Fields
    source
    ',4))]),t("details",q,[t("summary",null,[a[60]||(a[60]=t("a",{id:"IncompressibleNavierStokes.ode_method_cache",href:"#IncompressibleNavierStokes.ode_method_cache"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.ode_method_cache")],-1)),a[61]||(a[61]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[62]||(a[62]=e('
    julia
    ode_method_cache(method, setup, u, temp)
    Get time stepper cache for the given ODE method.
    source
    ',3))]),t("details",$,[t("summary",null,[a[63]||(a[63]=t("a",{id:"IncompressibleNavierStokes.runge_kutta_method",href:"#IncompressibleNavierStokes.runge_kutta_method"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.runge_kutta_method")],-1)),a[64]||(a[64]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[65]||(a[65]=e(`
    julia
    runge_kutta_method(
     A,
     b,
     c,
     r;
     T,
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod, IncompressibleNavierStokes.ImplicitRungeKuttaMethod}
    Get Runge Kutta method. The function checks whether the method is explicit.
    p_add_solve: whether to add a pressure solve step to the method.
    For implicit RK methods: newton_type, maxiter, abstol, reltol.
    source
    `,5))]),t("details",Y,[t("summary",null,[a[66]||(a[66]=t("a",{id:"IncompressibleNavierStokes.create_stepper",href:"#IncompressibleNavierStokes.create_stepper"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.create_stepper")],-1)),a[67]||(a[67]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[68]||(a[68]=e('
    julia
    create_stepper(method; setup, psolver, u, temp, t, n)
    Create time stepper.
    source
    ',3))]),t("details",_,[t("summary",null,[a[69]||(a[69]=t("a",{id:"IncompressibleNavierStokes.timestep",href:"#IncompressibleNavierStokes.timestep"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.timestep")],-1)),a[70]||(a[70]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[71]||(a[71]=e('
    julia
    timestep(method, stepper, Δt; θ = nothing)
    Perform one time step.
    Non-mutating/allocating/out-of-place version.
    See also timestep!.
    source
    ',5))]),t("details",t1,[t("summary",null,[a[72]||(a[72]=t("a",{id:"IncompressibleNavierStokes.timestep!",href:"#IncompressibleNavierStokes.timestep!"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.timestep!")],-1)),a[73]||(a[73]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[74]||(a[74]=e('
    julia
    timestep!(method, stepper, Δt; θ = nothing, cache)
    Perform one time step>
    Mutating/non-allocating/in-place version.
    See also timestep.
    source
    ',5))]),a[479]||(a[479]=t("h2",{id:"Adams-Bashforth-Crank-Nicolson-method",tabindex:"-1"},[s("Adams-Bashforth Crank-Nicolson method "),t("a",{class:"header-anchor",href:"#Adams-Bashforth-Crank-Nicolson-method","aria-label":'Permalink to "Adams-Bashforth Crank-Nicolson method {#Adams-Bashforth-Crank-Nicolson-method}"'},"​")],-1)),t("details",a1,[t("summary",null,[a[75]||(a[75]=t("a",{id:"IncompressibleNavierStokes.AdamsBashforthCrankNicolsonMethod",href:"#IncompressibleNavierStokes.AdamsBashforthCrankNicolsonMethod"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.AdamsBashforthCrankNicolsonMethod")],-1)),a[76]||(a[76]=s()),i(T,{type:"info",class:"jlObjectType jlType",text:"Type"})]),a[151]||(a[151]=e('
    julia
    struct AdamsBashforthCrankNicolsonMethod{T, M} <: IncompressibleNavierStokes.AbstractODEMethod{T}
    IMEX AB-CN: Adams-Bashforth for explicit convection (parameters α₁ and α₂) and Crank-Nicolson for implicit diffusion (implicitness θ). The method is second order for θ = 1/2.
    The LU decomposition of the LHS matrix is computed every time the time step changes.
    Note that, in contrast to explicit methods, the pressure from previous time steps has an influence on the accuracy of the velocity.
    ',4)),t("p",null,[a[91]||(a[91]=s("We here require that the time step ")),t("mjx-container",s1,[(l(),Q("svg",e1,a[77]||(a[77]=[e('',1)]))),a[78]||(a[78]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",{mathvariant:"normal"},"Δ"),t("mi",null,"t")])],-1))]),a[92]||(a[92]=s(" is constant. This methods uses Adams-Bashforth for the convective terms and Crank-Nicolson stepping for the diffusion and body force terms. Given the velocity field ")),t("mjx-container",Q1,[(l(),Q("svg",l1,a[79]||(a[79]=[e('',1)]))),a[80]||(a[80]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"u"),t("mn",null,"0")]),t("mo",null,"="),t("mi",null,"u"),t("mo",{stretchy:"false"},"("),t("msub",null,[t("mi",null,"t"),t("mn",null,"0")]),t("mo",{stretchy:"false"},")")])],-1))]),a[93]||(a[93]=s(" at a time ")),t("mjx-container",T1,[(l(),Q("svg",i1,a[81]||(a[81]=[e('',1)]))),a[82]||(a[82]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 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Assuming ")),t("mjx-container",I1,[(l(),Q("svg",F1,a[123]||(a[123]=[e('',1)]))),a[124]||(a[124]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",{mathvariant:"normal"},"Δ"),t("mi",null,"t")])],-1))]),a[128]||(a[128]=s(" is constant, we can precompute a Cholesky factorization of this matrix before starting time stepping."))]),t("p",null,[a[131]||(a[131]=s("We then compute the pressure difference ")),t("mjx-container",R1,[(l(),Q("svg",C1,a[129]||(a[129]=[e('',1)]))),a[130]||(a[130]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 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However, since this pressure is reused in the next time step, we perform an additional pressure solve to avoid accumulating first order errors. The resulting pressure ")),t("mjx-container",J1,[(l(),Q("svg",U1,a[143]||(a[143]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D45D",d:"M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z",style:{"stroke-width":"3"}})])])],-1)]))),a[144]||(a[144]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"p")])],-1))]),a[149]||(a[149]=s(" is then accurate to the same order as ")),t("mjx-container",X1,[(l(),Q("svg",W1,a[145]||(a[145]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D462",d:"M21 287Q21 295 30 318T55 370T99 420T158 442Q204 442 227 417T250 358Q250 340 216 246T182 105Q182 62 196 45T238 27T291 44T328 78L339 95Q341 99 377 247Q407 367 413 387T427 416Q444 431 463 431Q480 431 488 421T496 402L420 84Q419 79 419 68Q419 43 426 35T447 26Q469 29 482 57T512 145Q514 153 532 153Q551 153 551 144Q550 139 549 130T540 98T523 55T498 17T462 -8Q454 -10 438 -10Q372 -10 347 46Q345 45 336 36T318 21T296 6T267 -6T233 -11Q189 -11 155 7Q103 38 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z",style:{"stroke-width":"3"}})])])],-1)]))),a[146]||(a[146]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"u")])],-1))]),a[150]||(a[150]=s("."))]),a[152]||(a[152]=e('
    Fields
    • α₁
    • α₂
    • θ
    • p_add_solve
    • method_startup
    source
    ',3))]),a[480]||(a[480]=t("h2",{id:"One-leg-beta-method",tabindex:"-1"},[s("One-leg beta method "),t("a",{class:"header-anchor",href:"#One-leg-beta-method","aria-label":'Permalink to "One-leg beta method {#One-leg-beta-method}"'},"​")],-1)),t("details",q1,[t("summary",null,[a[153]||(a[153]=t("a",{id:"IncompressibleNavierStokes.OneLegMethod",href:"#IncompressibleNavierStokes.OneLegMethod"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.OneLegMethod")],-1)),a[154]||(a[154]=s()),i(T,{type:"info",class:"jlObjectType jlType",text:"Type"})]),a[202]||(a[202]=e('
    julia
    struct OneLegMethod{T, M} <: IncompressibleNavierStokes.AbstractODEMethod{T}
    Explicit one-leg β-method following symmetry-preserving discretization of turbulent flow. See Verstappen and Veldman [14] [16] for details.
    ',2)),t("p",null,[a[175]||(a[175]=s("We here require that the time step ")),t("mjx-container",$1,[(l(),Q("svg",Y1,a[155]||(a[155]=[e('',1)]))),a[156]||(a[156]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",{mathvariant:"normal"},"Δ"),t("mi",null,"t")])],-1))]),a[176]||(a[176]=s(" is constant. Given the velocity ")),t("mjx-container",_1,[(l(),Q("svg",t2,a[157]||(a[157]=[e('',1)]))),a[158]||(a[158]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"u"),t("mn",null,"0")])])],-1))]),a[177]||(a[177]=s(" and pressure ")),t("mjx-container",a2,[(l(),Q("svg",s2,a[159]||(a[159]=[e('',1)]))),a[160]||(a[160]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 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0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"t"),t("mn",null,"0")])])],-1))]),a[179]||(a[179]=s(" and their previous values ")),t("mjx-container",l2,[(l(),Q("svg",T2,a[163]||(a[163]=[e('',1)]))),a[164]||(a[164]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"u"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mo",null,"−"),t("mn",null,"1")])])])],-1))]),a[180]||(a[180]=s(" and ")),t("mjx-container",i2,[(l(),Q("svg",n2,a[165]||(a[165]=[e('',1)]))),a[166]||(a[166]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"p"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mo",null,"−"),t("mn",null,"1")])])])],-1))]),a[181]||(a[181]=s(" at the time ")),t("mjx-container",o2,[(l(),Q("svg",r2,a[167]||(a[167]=[e('',1)]))),a[168]||(a[168]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 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    julia
    abstract type AbstractRungeKuttaMethod{T} <: IncompressibleNavierStokes.AbstractODEMethod{T}
    Abstract Runge Kutta method.
    Fields
    source
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    julia
    struct ExplicitRungeKuttaMethod{T} <: IncompressibleNavierStokes.AbstractRungeKuttaMethod{T}
    Explicit Runge Kutta method. See Sanderse [17].
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We will now perform one time step to ")),t("mjx-container",F2,[(l(),Q("svg",R2,a[217]||(a[217]=[e('',1)]))),a[218]||(a[218]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"t"),t("mo",null,"="),t("msub",null,[t("mi",null,"t"),t("mn",null,"0")]),t("mo",null,"+"),t("mi",{mathvariant:"normal"},"Δ"),t("mi",null,"t")])],-1))]),a[232]||(a[232]=s(". 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385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z",style:{"stroke-width":"3"}})])])],-1)]))),a[278]||(a[278]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"u")])],-1))]),a[296]||(a[296]=s(" by doing an additional pressure projection after each outer time step ")),t("mjx-container",v3,[(l(),Q("svg",E3,a[279]||(a[279]=[e('',1)]))),a[280]||(a[280]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",{mathvariant:"normal"},"Δ"),t("mi",null,"t")])],-1))]),a[297]||(a[297]=s(" (if we know ")),t("mjx-container",Z3,[(l(),Q("svg",j3,a[281]||(a[281]=[e('',1)]))),a[282]||(a[282]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mfrac",null,[t("mrow",null,[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"normal"},"d")]),t("msub",null,[t("mi",null,"y"),t("mi",null,"M")])]),t("mrow",null,[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"normal"},"d")]),t("mi",null,"t")])]),t("mo",{stretchy:"false"},"("),t("mi",null,"t"),t("mo",{stretchy:"false"},")")])],-1))]),a[298]||(a[298]=s("), or to first order accuracy by simply returning ")),t("mjx-container",D3,[(l(),Q("svg",S3,a[283]||(a[283]=[e('',1)]))),a[284]||(a[284]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"p"),t("mi",null,"s")])])],-1))]),a[299]||(a[299]=s("."))]),t("p",null,[a[308]||(a[308]=s("Note that each of the sub-step velocities ")),t("mjx-container",V3,[(l(),Q("svg",I3,a[300]||(a[300]=[e('',1)]))),a[301]||(a[301]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"u"),t("mi",null,"i")])])],-1))]),a[309]||(a[309]=s(" is divergence free, after projecting the tentative velocities ")),t("mjx-container",F3,[(l(),Q("svg",R3,a[302]||(a[302]=[e('',1)]))),a[303]||(a[303]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"v"),t("mi",null,"i")])])],-1))]),a[310]||(a[310]=s(". This is ensured due to the judiciously chosen replacement of ")),t("mjx-container",C3,[(l(),Q("svg",N3,a[304]||(a[304]=[e('',1)]))),a[305]||(a[305]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mfrac",null,[t("mrow",null,[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"normal"},"d")]),t("msub",null,[t("mi",null,"y"),t("mi",null,"M")])]),t("mrow",null,[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"normal"},"d")]),t("mi",null,"t")])]),t("mo",{stretchy:"false"},"("),t("msub",null,[t("mi",null,"t"),t("mi",null,"i")]),t("mo",{stretchy:"false"},")")])],-1))]),a[311]||(a[311]=s(" with ")),t("mjx-container",K3,[(l(),Q("svg",A3,a[306]||(a[306]=[e('',1)]))),a[307]||(a[307]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mo",{stretchy:"false"},"("),t("msub",null,[t("mi",null,"y"),t("mi",null,"M")]),t("mo",{stretchy:"false"},"("),t("msub",null,[t("mi",null,"t"),t("mi",null,"i")]),t("mo",{stretchy:"false"},")"),t("mo",null,"−"),t("msub",null,[t("mi",null,"y"),t("mi",null,"M")]),t("mo",{stretchy:"false"},"("),t("msub",null,[t("mi",null,"t"),t("mn",null,"0")]),t("mo",{stretchy:"false"},")"),t("mo",{stretchy:"false"},")"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mo",null,"/")]),t("mi",{mathvariant:"normal"},"Δ"),t("msub",null,[t("mi",null,"t"),t("mi",null,"i")])])],-1))]),a[312]||(a[312]=s(". The space-discrete divergence-freeness is thus perfectly preserved, even though the time discretization introduces other errors."))]),a[314]||(a[314]=e('
    Fields
    • A
    • b
    • c
    • r
    • p_add_solve
    source
    ',3))]),t("details",B3,[t("summary",null,[a[315]||(a[315]=t("a",{id:"IncompressibleNavierStokes.ImplicitRungeKuttaMethod",href:"#IncompressibleNavierStokes.ImplicitRungeKuttaMethod"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.ImplicitRungeKuttaMethod")],-1)),a[316]||(a[316]=s()),i(T,{type:"info",class:"jlObjectType jlType",text:"Type"})]),a[317]||(a[317]=e('
    julia
    struct ImplicitRungeKuttaMethod{T} <: IncompressibleNavierStokes.AbstractRungeKuttaMethod{T}
    Implicit Runge Kutta method. See Sanderse [18].
    The implicit linear system is solved at each time step using Newton's method. The newton_type may be one of the following:
    • :no: Replace iteration matrix with I/Δt (no Jacobian)
    • :approximate: Build Jacobian once before iterations only
    • :full: Build Jacobian at each iteration
    Fields
    • A
    • b
    • c
    • r
    • newton_type
    • maxiter
    • abstol
    • reltol
    • p_add_solve
    source
    ',7))]),t("details",O3,[t("summary",null,[a[318]||(a[318]=t("a",{id:"IncompressibleNavierStokes.RKMethods",href:"#IncompressibleNavierStokes.RKMethods"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods")],-1)),a[319]||(a[319]=s()),i(T,{type:"info",class:"jlObjectType jlModule",text:"Module"})]),a[320]||(a[320]=e('
    Set up Butcher arrays A, b, and c, as well as and SSP coefficient r. For families of methods, optional input s is the number of stages.
    Original (MATLAB) by David Ketcheson, extended by Benjamin Sanderse.
    Exports
    source
    ',5))]),t("details",P3,[t("summary",null,[a[321]||(a[321]=t("a",{id:"IncompressibleNavierStokes.LMWray3",href:"#IncompressibleNavierStokes.LMWray3"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.LMWray3")],-1)),a[322]||(a[322]=s()),i(T,{type:"info",class:"jlObjectType jlType",text:"Type"})]),a[323]||(a[323]=e('
    julia
    struct LMWray3{T} <: IncompressibleNavierStokes.AbstractRungeKuttaMethod{T}
    Low memory Wray 3rd order scheme. Uses 3 vector fields and one scalar field.
    Fields
    source
    ',4))]),a[482]||(a[482]=t("h3",{id:"Explicit-Methods",tabindex:"-1"},[s("Explicit Methods "),t("a",{class:"header-anchor",href:"#Explicit-Methods","aria-label":'Permalink to "Explicit Methods {#Explicit-Methods}"'},"​")],-1)),t("details",G3,[t("summary",null,[a[324]||(a[324]=t("a",{id:"IncompressibleNavierStokes.RKMethods.FE11",href:"#IncompressibleNavierStokes.RKMethods.FE11"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.FE11")],-1)),a[325]||(a[325]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[326]||(a[326]=e(`
    julia
    FE11(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod, IncompressibleNavierStokes.ImplicitRungeKuttaMethod}
    Get Runge Kutta method. The function checks whether the method is explicit.
    p_add_solve: whether to add a pressure solve step to the method.
    For implicit RK methods: newton_type, maxiter, abstol, reltol.
    source
    `,5))]),t("details",Y,[t("summary",null,[a[66]||(a[66]=t("a",{id:"IncompressibleNavierStokes.create_stepper",href:"#IncompressibleNavierStokes.create_stepper"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.create_stepper")],-1)),a[67]||(a[67]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[68]||(a[68]=e('
    julia
    create_stepper(method; setup, psolver, u, temp, t, n)
    Create time stepper.
    source
    ',3))]),t("details",_,[t("summary",null,[a[69]||(a[69]=t("a",{id:"IncompressibleNavierStokes.timestep",href:"#IncompressibleNavierStokes.timestep"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.timestep")],-1)),a[70]||(a[70]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[71]||(a[71]=e('
    julia
    timestep(method, stepper, Δt; θ = nothing)
    Perform one time step.
    Non-mutating/allocating/out-of-place version.
    See also timestep!.
    source
    ',5))]),t("details",t1,[t("summary",null,[a[72]||(a[72]=t("a",{id:"IncompressibleNavierStokes.timestep!",href:"#IncompressibleNavierStokes.timestep!"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.timestep!")],-1)),a[73]||(a[73]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[74]||(a[74]=e('
    julia
    timestep!(method, stepper, Δt; θ = nothing, cache)
    Perform one time step>
    Mutating/non-allocating/in-place version.
    See also timestep.
    source
    ',5))]),a[479]||(a[479]=t("h2",{id:"Adams-Bashforth-Crank-Nicolson-method",tabindex:"-1"},[s("Adams-Bashforth Crank-Nicolson method "),t("a",{class:"header-anchor",href:"#Adams-Bashforth-Crank-Nicolson-method","aria-label":'Permalink to "Adams-Bashforth Crank-Nicolson method {#Adams-Bashforth-Crank-Nicolson-method}"'},"​")],-1)),t("details",a1,[t("summary",null,[a[75]||(a[75]=t("a",{id:"IncompressibleNavierStokes.AdamsBashforthCrankNicolsonMethod",href:"#IncompressibleNavierStokes.AdamsBashforthCrankNicolsonMethod"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.AdamsBashforthCrankNicolsonMethod")],-1)),a[76]||(a[76]=s()),i(T,{type:"info",class:"jlObjectType jlType",text:"Type"})]),a[151]||(a[151]=e('
    julia
    struct AdamsBashforthCrankNicolsonMethod{T, M} <: IncompressibleNavierStokes.AbstractODEMethod{T}
    IMEX AB-CN: Adams-Bashforth for explicit convection (parameters α₁ and α₂) and Crank-Nicolson for implicit diffusion (implicitness θ). The method is second order for θ = 1/2.
    The LU decomposition of the LHS matrix is computed every time the time step changes.
    Note that, in contrast to explicit methods, the pressure from previous time steps has an influence on the accuracy of the velocity.
    ',4)),t("p",null,[a[91]||(a[91]=s("We here require that the time step ")),t("mjx-container",s1,[(l(),Q("svg",e1,a[77]||(a[77]=[e('',1)]))),a[78]||(a[78]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",{mathvariant:"normal"},"Δ"),t("mi",null,"t")])],-1))]),a[92]||(a[92]=s(" is constant. This methods uses Adams-Bashforth for the convective terms and Crank-Nicolson stepping for the diffusion and body force terms. Given the velocity field ")),t("mjx-container",Q1,[(l(),Q("svg",l1,a[79]||(a[79]=[e('',1)]))),a[80]||(a[80]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"u"),t("mn",null,"0")]),t("mo",null,"="),t("mi",null,"u"),t("mo",{stretchy:"false"},"("),t("msub",null,[t("mi",null,"t"),t("mn",null,"0")]),t("mo",{stretchy:"false"},")")])],-1))]),a[93]||(a[93]=s(" at a time ")),t("mjx-container",T1,[(l(),Q("svg",i1,a[81]||(a[81]=[e('',1)]))),a[82]||(a[82]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"t"),t("mn",null,"0")])])],-1))]),a[94]||(a[94]=s(" and its previous value ")),t("mjx-container",n1,[(l(),Q("svg",o1,a[83]||(a[83]=[e('',1)]))),a[84]||(a[84]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"u"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mo",null,"−"),t("mn",null,"1")])]),t("mo",null,"="),t("mi",null,"u"),t("mo",{stretchy:"false"},"("),t("msub",null,[t("mi",null,"t"),t("mn",null,"0")]),t("mo",null,"−"),t("mi",{mathvariant:"normal"},"Δ"),t("mi",null,"t"),t("mo",{stretchy:"false"},")")])],-1))]),a[95]||(a[95]=s(" at the previous time ")),t("mjx-container",r1,[(l(),Q("svg",d1,a[85]||(a[85]=[e('',1)]))),a[86]||(a[86]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 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However, since this pressure is reused in the next time step, we perform an additional pressure solve to avoid accumulating first order errors. The resulting pressure ")),t("mjx-container",J1,[(l(),Q("svg",U1,a[143]||(a[143]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D45D",d:"M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z",style:{"stroke-width":"3"}})])])],-1)]))),a[144]||(a[144]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"p")])],-1))]),a[149]||(a[149]=s(" is then accurate to the same order as ")),t("mjx-container",X1,[(l(),Q("svg",W1,a[145]||(a[145]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D462",d:"M21 287Q21 295 30 318T55 370T99 420T158 442Q204 442 227 417T250 358Q250 340 216 246T182 105Q182 62 196 45T238 27T291 44T328 78L339 95Q341 99 377 247Q407 367 413 387T427 416Q444 431 463 431Q480 431 488 421T496 402L420 84Q419 79 419 68Q419 43 426 35T447 26Q469 29 482 57T512 145Q514 153 532 153Q551 153 551 144Q550 139 549 130T540 98T523 55T498 17T462 -8Q454 -10 438 -10Q372 -10 347 46Q345 45 336 36T318 21T296 6T267 -6T233 -11Q189 -11 155 7Q103 38 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z",style:{"stroke-width":"3"}})])])],-1)]))),a[146]||(a[146]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"u")])],-1))]),a[150]||(a[150]=s("."))]),a[152]||(a[152]=e('
    Fields
    • α₁
    • α₂
    • θ
    • p_add_solve
    • method_startup
    source
    ',3))]),a[480]||(a[480]=t("h2",{id:"One-leg-beta-method",tabindex:"-1"},[s("One-leg beta method "),t("a",{class:"header-anchor",href:"#One-leg-beta-method","aria-label":'Permalink to "One-leg beta method {#One-leg-beta-method}"'},"​")],-1)),t("details",q1,[t("summary",null,[a[153]||(a[153]=t("a",{id:"IncompressibleNavierStokes.OneLegMethod",href:"#IncompressibleNavierStokes.OneLegMethod"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.OneLegMethod")],-1)),a[154]||(a[154]=s()),i(T,{type:"info",class:"jlObjectType jlType",text:"Type"})]),a[202]||(a[202]=e('
    julia
    struct OneLegMethod{T, M} <: IncompressibleNavierStokes.AbstractODEMethod{T}
    Explicit one-leg β-method following symmetry-preserving discretization of turbulent flow. See Verstappen and Veldman [14] [16] for details.
    ',2)),t("p",null,[a[175]||(a[175]=s("We here require that the time step ")),t("mjx-container",$1,[(l(),Q("svg",Y1,a[155]||(a[155]=[e('',1)]))),a[156]||(a[156]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",{mathvariant:"normal"},"Δ"),t("mi",null,"t")])],-1))]),a[176]||(a[176]=s(" is constant. Given the velocity ")),t("mjx-container",_1,[(l(),Q("svg",t2,a[157]||(a[157]=[e('',1)]))),a[158]||(a[158]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"u"),t("mn",null,"0")])])],-1))]),a[177]||(a[177]=s(" and pressure ")),t("mjx-container",a2,[(l(),Q("svg",s2,a[159]||(a[159]=[e('',1)]))),a[160]||(a[160]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 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0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"t"),t("mn",null,"0")])])],-1))]),a[179]||(a[179]=s(" and their previous values ")),t("mjx-container",l2,[(l(),Q("svg",T2,a[163]||(a[163]=[e('',1)]))),a[164]||(a[164]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"u"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mo",null,"−"),t("mn",null,"1")])])])],-1))]),a[180]||(a[180]=s(" and ")),t("mjx-container",i2,[(l(),Q("svg",n2,a[165]||(a[165]=[e('',1)]))),a[166]||(a[166]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"p"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mo",null,"−"),t("mn",null,"1")])])])],-1))]),a[181]||(a[181]=s(" at the time ")),t("mjx-container",o2,[(l(),Q("svg",r2,a[167]||(a[167]=[e('',1)]))),a[168]||(a[168]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 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    julia
    abstract type AbstractRungeKuttaMethod{T} <: IncompressibleNavierStokes.AbstractODEMethod{T}
    Abstract Runge Kutta method.
    Fields
    source
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    julia
    struct ExplicitRungeKuttaMethod{T} <: IncompressibleNavierStokes.AbstractRungeKuttaMethod{T}
    Explicit Runge Kutta method. See Sanderse [17].
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We will now perform one time step to ")),t("mjx-container",F2,[(l(),Q("svg",R2,a[217]||(a[217]=[e('',1)]))),a[218]||(a[218]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"t"),t("mo",null,"="),t("msub",null,[t("mi",null,"t"),t("mn",null,"0")]),t("mo",null,"+"),t("mi",{mathvariant:"normal"},"Δ"),t("mi",null,"t")])],-1))]),a[232]||(a[232]=s(". 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")),t("mjx-container",F3,[(l(),Q("svg",R3,a[302]||(a[302]=[e('',1)]))),a[303]||(a[303]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"v"),t("mi",null,"i")])])],-1))]),a[310]||(a[310]=s(". This is ensured due to the judiciously chosen replacement of ")),t("mjx-container",C3,[(l(),Q("svg",N3,a[304]||(a[304]=[e('',1)]))),a[305]||(a[305]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mfrac",null,[t("mrow",null,[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"normal"},"d")]),t("msub",null,[t("mi",null,"y"),t("mi",null,"M")])]),t("mrow",null,[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"normal"},"d")]),t("mi",null,"t")])]),t("mo",{stretchy:"false"},"("),t("msub",null,[t("mi",null,"t"),t("mi",null,"i")]),t("mo",{stretchy:"false"},")")])],-1))]),a[311]||(a[311]=s(" with ")),t("mjx-container",K3,[(l(),Q("svg",A3,a[306]||(a[306]=[e('',1)]))),a[307]||(a[307]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mo",{stretchy:"false"},"("),t("msub",null,[t("mi",null,"y"),t("mi",null,"M")]),t("mo",{stretchy:"false"},"("),t("msub",null,[t("mi",null,"t"),t("mi",null,"i")]),t("mo",{stretchy:"false"},")"),t("mo",null,"−"),t("msub",null,[t("mi",null,"y"),t("mi",null,"M")]),t("mo",{stretchy:"false"},"("),t("msub",null,[t("mi",null,"t"),t("mn",null,"0")]),t("mo",{stretchy:"false"},")"),t("mo",{stretchy:"false"},")"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mo",null,"/")]),t("mi",{mathvariant:"normal"},"Δ"),t("msub",null,[t("mi",null,"t"),t("mi",null,"i")])])],-1))]),a[312]||(a[312]=s(". The space-discrete divergence-freeness is thus perfectly preserved, even though the time discretization introduces other errors."))]),a[314]||(a[314]=e('
    Fields
    • A
    • b
    • c
    • r
    • p_add_solve
    source
    ',3))]),t("details",B3,[t("summary",null,[a[315]||(a[315]=t("a",{id:"IncompressibleNavierStokes.ImplicitRungeKuttaMethod",href:"#IncompressibleNavierStokes.ImplicitRungeKuttaMethod"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.ImplicitRungeKuttaMethod")],-1)),a[316]||(a[316]=s()),i(T,{type:"info",class:"jlObjectType jlType",text:"Type"})]),a[317]||(a[317]=e('
    julia
    struct ImplicitRungeKuttaMethod{T} <: IncompressibleNavierStokes.AbstractRungeKuttaMethod{T}
    Implicit Runge Kutta method. See Sanderse [18].
    The implicit linear system is solved at each time step using Newton's method. The newton_type may be one of the following:
    • :no: Replace iteration matrix with I/Δt (no Jacobian)
    • :approximate: Build Jacobian once before iterations only
    • :full: Build Jacobian at each iteration
    Fields
    • A
    • b
    • c
    • r
    • newton_type
    • maxiter
    • abstol
    • reltol
    • p_add_solve
    source
    ',7))]),t("details",O3,[t("summary",null,[a[318]||(a[318]=t("a",{id:"IncompressibleNavierStokes.RKMethods",href:"#IncompressibleNavierStokes.RKMethods"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods")],-1)),a[319]||(a[319]=s()),i(T,{type:"info",class:"jlObjectType jlModule",text:"Module"})]),a[320]||(a[320]=e('
    Set up Butcher arrays A, b, and c, as well as and SSP coefficient r. For families of methods, optional input s is the number of stages.
    Original (MATLAB) by David Ketcheson, extended by Benjamin Sanderse.
    Exports
    source
    ',5))]),t("details",P3,[t("summary",null,[a[321]||(a[321]=t("a",{id:"IncompressibleNavierStokes.LMWray3",href:"#IncompressibleNavierStokes.LMWray3"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.LMWray3")],-1)),a[322]||(a[322]=s()),i(T,{type:"info",class:"jlObjectType jlType",text:"Type"})]),a[323]||(a[323]=e('
    julia
    struct LMWray3{T} <: IncompressibleNavierStokes.AbstractRungeKuttaMethod{T}
    Low memory Wray 3rd order scheme. Uses 3 vector fields and one scalar field.
    Fields
    source
    ',4))]),a[482]||(a[482]=t("h3",{id:"Explicit-Methods",tabindex:"-1"},[s("Explicit Methods "),t("a",{class:"header-anchor",href:"#Explicit-Methods","aria-label":'Permalink to "Explicit Methods {#Explicit-Methods}"'},"​")],-1)),t("details",G3,[t("summary",null,[a[324]||(a[324]=t("a",{id:"IncompressibleNavierStokes.RKMethods.FE11",href:"#IncompressibleNavierStokes.RKMethods.FE11"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.FE11")],-1)),a[325]||(a[325]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[326]||(a[326]=e(`
    julia
    FE11(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    FE11 (Forward Euler).
    source
    `,3))]),t("details",z3,[t("summary",null,[a[327]||(a[327]=t("a",{id:"IncompressibleNavierStokes.RKMethods.SSP22",href:"#IncompressibleNavierStokes.RKMethods.SSP22"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.SSP22")],-1)),a[328]||(a[328]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[329]||(a[329]=e(`
    julia
    SSP22(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    FE11 (Forward Euler).
    source
    `,3))]),t("details",z3,[t("summary",null,[a[327]||(a[327]=t("a",{id:"IncompressibleNavierStokes.RKMethods.SSP22",href:"#IncompressibleNavierStokes.RKMethods.SSP22"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.SSP22")],-1)),a[328]||(a[328]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[329]||(a[329]=e(`
    julia
    SSP22(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    SSP22.
    source
    `,3))]),t("details",J3,[t("summary",null,[a[330]||(a[330]=t("a",{id:"IncompressibleNavierStokes.RKMethods.SSP42",href:"#IncompressibleNavierStokes.RKMethods.SSP42"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.SSP42")],-1)),a[331]||(a[331]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[332]||(a[332]=e(`
    julia
    SSP42(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    SSP22.
    source
    `,3))]),t("details",J3,[t("summary",null,[a[330]||(a[330]=t("a",{id:"IncompressibleNavierStokes.RKMethods.SSP42",href:"#IncompressibleNavierStokes.RKMethods.SSP42"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.SSP42")],-1)),a[331]||(a[331]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[332]||(a[332]=e(`
    julia
    SSP42(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    SSP42.
    source
    `,3))]),t("details",U3,[t("summary",null,[a[333]||(a[333]=t("a",{id:"IncompressibleNavierStokes.RKMethods.SSP33",href:"#IncompressibleNavierStokes.RKMethods.SSP33"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.SSP33")],-1)),a[334]||(a[334]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[335]||(a[335]=e(`
    julia
    SSP33(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    SSP42.
    source
    `,3))]),t("details",U3,[t("summary",null,[a[333]||(a[333]=t("a",{id:"IncompressibleNavierStokes.RKMethods.SSP33",href:"#IncompressibleNavierStokes.RKMethods.SSP33"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.SSP33")],-1)),a[334]||(a[334]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[335]||(a[335]=e(`
    julia
    SSP33(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    SSP33.
    source
    `,3))]),t("details",X3,[t("summary",null,[a[336]||(a[336]=t("a",{id:"IncompressibleNavierStokes.RKMethods.SSP43",href:"#IncompressibleNavierStokes.RKMethods.SSP43"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.SSP43")],-1)),a[337]||(a[337]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[338]||(a[338]=e(`
    julia
    SSP43(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    SSP33.
    source
    `,3))]),t("details",X3,[t("summary",null,[a[336]||(a[336]=t("a",{id:"IncompressibleNavierStokes.RKMethods.SSP43",href:"#IncompressibleNavierStokes.RKMethods.SSP43"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.SSP43")],-1)),a[337]||(a[337]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[338]||(a[338]=e(`
    julia
    SSP43(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    SSP43.
    source
    `,3))]),t("details",W3,[t("summary",null,[a[339]||(a[339]=t("a",{id:"IncompressibleNavierStokes.RKMethods.SSP104",href:"#IncompressibleNavierStokes.RKMethods.SSP104"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.SSP104")],-1)),a[340]||(a[340]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[341]||(a[341]=e(`
    julia
    SSP104(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    SSP43.
    source
    `,3))]),t("details",W3,[t("summary",null,[a[339]||(a[339]=t("a",{id:"IncompressibleNavierStokes.RKMethods.SSP104",href:"#IncompressibleNavierStokes.RKMethods.SSP104"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.SSP104")],-1)),a[340]||(a[340]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[341]||(a[341]=e(`
    julia
    SSP104(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    SSP104.
    source
    `,3))]),t("details",q3,[t("summary",null,[a[342]||(a[342]=t("a",{id:"IncompressibleNavierStokes.RKMethods.rSSPs2",href:"#IncompressibleNavierStokes.RKMethods.rSSPs2"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.rSSPs2")],-1)),a[343]||(a[343]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[344]||(a[344]=e(`
    julia
    rSSPs2(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    SSP104.
    source
    `,3))]),t("details",q3,[t("summary",null,[a[342]||(a[342]=t("a",{id:"IncompressibleNavierStokes.RKMethods.rSSPs2",href:"#IncompressibleNavierStokes.RKMethods.rSSPs2"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.rSSPs2")],-1)),a[343]||(a[343]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[344]||(a[344]=e(`
    julia
    rSSPs2(
 ;
     ...
 ) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod, IncompressibleNavierStokes.ImplicitRungeKuttaMethod}
 rSSPs2(
     s;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod, IncompressibleNavierStokes.ImplicitRungeKuttaMethod}
    Rational (optimal, low-storage) s-stage 2nd order SSP.
    source
    `,3))]),t("details",$3,[t("summary",null,[a[345]||(a[345]=t("a",{id:"IncompressibleNavierStokes.RKMethods.rSSPs3",href:"#IncompressibleNavierStokes.RKMethods.rSSPs3"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.rSSPs3")],-1)),a[346]||(a[346]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[347]||(a[347]=e(`
    julia
    rSSPs3(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod, IncompressibleNavierStokes.ImplicitRungeKuttaMethod}
    Rational (optimal, low-storage) s-stage 2nd order SSP.
    source
    `,3))]),t("details",$3,[t("summary",null,[a[345]||(a[345]=t("a",{id:"IncompressibleNavierStokes.RKMethods.rSSPs3",href:"#IncompressibleNavierStokes.RKMethods.rSSPs3"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.rSSPs3")],-1)),a[346]||(a[346]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[347]||(a[347]=e(`
    julia
    rSSPs3(
 ;
     ...
 ) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod, IncompressibleNavierStokes.ImplicitRungeKuttaMethod}
 rSSPs3(
     s;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod, IncompressibleNavierStokes.ImplicitRungeKuttaMethod}
    Rational (optimal, low-storage) s^2-stage 3rd order SSP.
    source
    `,3))]),t("details",Y3,[t("summary",null,[a[348]||(a[348]=t("a",{id:"IncompressibleNavierStokes.RKMethods.Wray3",href:"#IncompressibleNavierStokes.RKMethods.Wray3"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.Wray3")],-1)),a[349]||(a[349]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[350]||(a[350]=e(`
    julia
    Wray3(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod, IncompressibleNavierStokes.ImplicitRungeKuttaMethod}
    Rational (optimal, low-storage) s^2-stage 3rd order SSP.
    source
    `,3))]),t("details",Y3,[t("summary",null,[a[348]||(a[348]=t("a",{id:"IncompressibleNavierStokes.RKMethods.Wray3",href:"#IncompressibleNavierStokes.RKMethods.Wray3"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.Wray3")],-1)),a[349]||(a[349]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[350]||(a[350]=e(`
    julia
    Wray3(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    Wray's RK3.
    source
    `,3))]),t("details",_3,[t("summary",null,[a[351]||(a[351]=t("a",{id:"IncompressibleNavierStokes.RKMethods.RK56",href:"#IncompressibleNavierStokes.RKMethods.RK56"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.RK56")],-1)),a[352]||(a[352]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[353]||(a[353]=e(`
    julia
    RK56(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    Wray's RK3.
    source
    `,3))]),t("details",_3,[t("summary",null,[a[351]||(a[351]=t("a",{id:"IncompressibleNavierStokes.RKMethods.RK56",href:"#IncompressibleNavierStokes.RKMethods.RK56"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.RK56")],-1)),a[352]||(a[352]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[353]||(a[353]=e(`
    julia
    RK56(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    RK56.
    source
    `,3))]),t("details",t4,[t("summary",null,[a[354]||(a[354]=t("a",{id:"IncompressibleNavierStokes.RKMethods.DOPRI6",href:"#IncompressibleNavierStokes.RKMethods.DOPRI6"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.DOPRI6")],-1)),a[355]||(a[355]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[356]||(a[356]=e(`
    julia
    DOPRI6(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    RK56.
    source
    `,3))]),t("details",t4,[t("summary",null,[a[354]||(a[354]=t("a",{id:"IncompressibleNavierStokes.RKMethods.DOPRI6",href:"#IncompressibleNavierStokes.RKMethods.DOPRI6"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.DOPRI6")],-1)),a[355]||(a[355]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[356]||(a[356]=e(`
    julia
    DOPRI6(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    Dormand-Price pair.
    source
    `,3))]),a[483]||(a[483]=t("h3",{id:"Implicit-Methods",tabindex:"-1"},[s("Implicit Methods "),t("a",{class:"header-anchor",href:"#Implicit-Methods","aria-label":'Permalink to "Implicit Methods {#Implicit-Methods}"'},"​")],-1)),t("details",a4,[t("summary",null,[a[357]||(a[357]=t("a",{id:"IncompressibleNavierStokes.RKMethods.BE11",href:"#IncompressibleNavierStokes.RKMethods.BE11"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.BE11")],-1)),a[358]||(a[358]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[359]||(a[359]=e(`
    julia
    BE11(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    Dormand-Price pair.
    source
    `,3))]),a[483]||(a[483]=t("h3",{id:"Implicit-Methods",tabindex:"-1"},[s("Implicit Methods "),t("a",{class:"header-anchor",href:"#Implicit-Methods","aria-label":'Permalink to "Implicit Methods {#Implicit-Methods}"'},"​")],-1)),t("details",a4,[t("summary",null,[a[357]||(a[357]=t("a",{id:"IncompressibleNavierStokes.RKMethods.BE11",href:"#IncompressibleNavierStokes.RKMethods.BE11"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.BE11")],-1)),a[358]||(a[358]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[359]||(a[359]=e(`
    julia
    BE11(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    Backward Euler.
    source
    `,3))]),t("details",s4,[t("summary",null,[a[360]||(a[360]=t("a",{id:"IncompressibleNavierStokes.RKMethods.SDIRK34",href:"#IncompressibleNavierStokes.RKMethods.SDIRK34"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.SDIRK34")],-1)),a[361]||(a[361]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[362]||(a[362]=e(`
    julia
    SDIRK34(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    Backward Euler.
    source
    `,3))]),t("details",s4,[t("summary",null,[a[360]||(a[360]=t("a",{id:"IncompressibleNavierStokes.RKMethods.SDIRK34",href:"#IncompressibleNavierStokes.RKMethods.SDIRK34"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.SDIRK34")],-1)),a[361]||(a[361]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[362]||(a[362]=e(`
    julia
    SDIRK34(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    3-stage, 4th order singly diagonally implicit (SSP).
    source
    `,3))]),t("details",e4,[t("summary",null,[a[363]||(a[363]=t("a",{id:"IncompressibleNavierStokes.RKMethods.ISSPm2",href:"#IncompressibleNavierStokes.RKMethods.ISSPm2"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.ISSPm2")],-1)),a[364]||(a[364]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[365]||(a[365]=e(`
    julia
    ISSPm2(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    3-stage, 4th order singly diagonally implicit (SSP).
    source
    `,3))]),t("details",e4,[t("summary",null,[a[363]||(a[363]=t("a",{id:"IncompressibleNavierStokes.RKMethods.ISSPm2",href:"#IncompressibleNavierStokes.RKMethods.ISSPm2"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.ISSPm2")],-1)),a[364]||(a[364]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[365]||(a[365]=e(`
    julia
    ISSPm2(
 ;
     ...
 ) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
 ISSPm2(
     s;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod, IncompressibleNavierStokes.ImplicitRungeKuttaMethod}
    Optimal DIRK SSP schemes of order 2.
    source
    `,3))]),t("details",Q4,[t("summary",null,[a[366]||(a[366]=t("a",{id:"IncompressibleNavierStokes.RKMethods.ISSPs3",href:"#IncompressibleNavierStokes.RKMethods.ISSPs3"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.ISSPs3")],-1)),a[367]||(a[367]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[368]||(a[368]=e(`
    julia
    ISSPs3(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod, IncompressibleNavierStokes.ImplicitRungeKuttaMethod}
    Optimal DIRK SSP schemes of order 2.
    source
    `,3))]),t("details",Q4,[t("summary",null,[a[366]||(a[366]=t("a",{id:"IncompressibleNavierStokes.RKMethods.ISSPs3",href:"#IncompressibleNavierStokes.RKMethods.ISSPs3"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.ISSPs3")],-1)),a[367]||(a[367]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[368]||(a[368]=e(`
    julia
    ISSPs3(
 ;
     ...
 ) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
 ISSPs3(
     s;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod, IncompressibleNavierStokes.ImplicitRungeKuttaMethod}
    Optimal DIRK SSP schemes of order 3.
    source
    `,3))]),a[484]||(a[484]=t("h3",{id:"Half-explicit-methods",tabindex:"-1"},[s("Half explicit methods "),t("a",{class:"header-anchor",href:"#Half-explicit-methods","aria-label":'Permalink to "Half explicit methods {#Half-explicit-methods}"'},"​")],-1)),t("details",l4,[t("summary",null,[a[369]||(a[369]=t("a",{id:"IncompressibleNavierStokes.RKMethods.HEM3",href:"#IncompressibleNavierStokes.RKMethods.HEM3"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.HEM3")],-1)),a[370]||(a[370]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[371]||(a[371]=e(`
    julia
    HEM3(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod, IncompressibleNavierStokes.ImplicitRungeKuttaMethod}
    Optimal DIRK SSP schemes of order 3.
    source
    `,3))]),a[484]||(a[484]=t("h3",{id:"Half-explicit-methods",tabindex:"-1"},[s("Half explicit methods "),t("a",{class:"header-anchor",href:"#Half-explicit-methods","aria-label":'Permalink to "Half explicit methods {#Half-explicit-methods}"'},"​")],-1)),t("details",l4,[t("summary",null,[a[369]||(a[369]=t("a",{id:"IncompressibleNavierStokes.RKMethods.HEM3",href:"#IncompressibleNavierStokes.RKMethods.HEM3"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.HEM3")],-1)),a[370]||(a[370]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[371]||(a[371]=e(`
    julia
    HEM3(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    Brasey and Hairer.
    source
    `,3))]),t("details",T4,[t("summary",null,[a[372]||(a[372]=t("a",{id:"IncompressibleNavierStokes.RKMethods.HEM3BS",href:"#IncompressibleNavierStokes.RKMethods.HEM3BS"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.HEM3BS")],-1)),a[373]||(a[373]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[374]||(a[374]=e(`
    julia
    HEM3BS(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    Brasey and Hairer.
    source
    `,3))]),t("details",T4,[t("summary",null,[a[372]||(a[372]=t("a",{id:"IncompressibleNavierStokes.RKMethods.HEM3BS",href:"#IncompressibleNavierStokes.RKMethods.HEM3BS"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.HEM3BS")],-1)),a[373]||(a[373]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[374]||(a[374]=e(`
    julia
    HEM3BS(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    HEM3BS.
    source
    `,3))]),t("details",i4,[t("summary",null,[a[375]||(a[375]=t("a",{id:"IncompressibleNavierStokes.RKMethods.HEM5",href:"#IncompressibleNavierStokes.RKMethods.HEM5"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.HEM5")],-1)),a[376]||(a[376]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[377]||(a[377]=e(`
    julia
    HEM5(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    HEM3BS.
    source
    `,3))]),t("details",i4,[t("summary",null,[a[375]||(a[375]=t("a",{id:"IncompressibleNavierStokes.RKMethods.HEM5",href:"#IncompressibleNavierStokes.RKMethods.HEM5"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.HEM5")],-1)),a[376]||(a[376]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[377]||(a[377]=e(`
    julia
    HEM5(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    Brasey and Hairer, 5 stage, 4th order.
    source
    `,3))]),a[485]||(a[485]=t("h3",{id:"Classical-Methods",tabindex:"-1"},[s("Classical Methods "),t("a",{class:"header-anchor",href:"#Classical-Methods","aria-label":'Permalink to "Classical Methods {#Classical-Methods}"'},"​")],-1)),t("details",n4,[t("summary",null,[a[378]||(a[378]=t("a",{id:"IncompressibleNavierStokes.RKMethods.GL1",href:"#IncompressibleNavierStokes.RKMethods.GL1"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.GL1")],-1)),a[379]||(a[379]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[380]||(a[380]=e(`
    julia
    GL1(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    Brasey and Hairer, 5 stage, 4th order.
    source
    `,3))]),a[485]||(a[485]=t("h3",{id:"Classical-Methods",tabindex:"-1"},[s("Classical Methods "),t("a",{class:"header-anchor",href:"#Classical-Methods","aria-label":'Permalink to "Classical Methods {#Classical-Methods}"'},"​")],-1)),t("details",n4,[t("summary",null,[a[378]||(a[378]=t("a",{id:"IncompressibleNavierStokes.RKMethods.GL1",href:"#IncompressibleNavierStokes.RKMethods.GL1"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.GL1")],-1)),a[379]||(a[379]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[380]||(a[380]=e(`
    julia
    GL1(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    GL1.
    source
    `,3))]),t("details",o4,[t("summary",null,[a[381]||(a[381]=t("a",{id:"IncompressibleNavierStokes.RKMethods.GL2",href:"#IncompressibleNavierStokes.RKMethods.GL2"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.GL2")],-1)),a[382]||(a[382]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[383]||(a[383]=e(`
    julia
    GL2(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    GL1.
    source
    `,3))]),t("details",o4,[t("summary",null,[a[381]||(a[381]=t("a",{id:"IncompressibleNavierStokes.RKMethods.GL2",href:"#IncompressibleNavierStokes.RKMethods.GL2"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.GL2")],-1)),a[382]||(a[382]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[383]||(a[383]=e(`
    julia
    GL2(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    GL2.
    source
    `,3))]),t("details",r4,[t("summary",null,[a[384]||(a[384]=t("a",{id:"IncompressibleNavierStokes.RKMethods.GL3",href:"#IncompressibleNavierStokes.RKMethods.GL3"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.GL3")],-1)),a[385]||(a[385]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[386]||(a[386]=e(`
    julia
    GL3(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    GL2.
    source
    `,3))]),t("details",r4,[t("summary",null,[a[384]||(a[384]=t("a",{id:"IncompressibleNavierStokes.RKMethods.GL3",href:"#IncompressibleNavierStokes.RKMethods.GL3"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.GL3")],-1)),a[385]||(a[385]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[386]||(a[386]=e(`
    julia
    GL3(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    GL3.
    source
    `,3))]),t("details",d4,[t("summary",null,[a[387]||(a[387]=t("a",{id:"IncompressibleNavierStokes.RKMethods.RIA1",href:"#IncompressibleNavierStokes.RKMethods.RIA1"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.RIA1")],-1)),a[388]||(a[388]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[389]||(a[389]=e(`
    julia
    RIA1(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    GL3.
    source
    `,3))]),t("details",d4,[t("summary",null,[a[387]||(a[387]=t("a",{id:"IncompressibleNavierStokes.RKMethods.RIA1",href:"#IncompressibleNavierStokes.RKMethods.RIA1"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.RIA1")],-1)),a[388]||(a[388]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[389]||(a[389]=e(`
    julia
    RIA1(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    This is implicit Euler.
    source
    `,3))]),t("details",m4,[t("summary",null,[a[390]||(a[390]=t("a",{id:"IncompressibleNavierStokes.RKMethods.RIA2",href:"#IncompressibleNavierStokes.RKMethods.RIA2"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.RIA2")],-1)),a[391]||(a[391]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[392]||(a[392]=e(`
    julia
    RIA2(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    This is implicit Euler.
    source
    `,3))]),t("details",m4,[t("summary",null,[a[390]||(a[390]=t("a",{id:"IncompressibleNavierStokes.RKMethods.RIA2",href:"#IncompressibleNavierStokes.RKMethods.RIA2"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.RIA2")],-1)),a[391]||(a[391]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[392]||(a[392]=e(`
    julia
    RIA2(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    RIA2.
    source
    `,3))]),t("details",p4,[t("summary",null,[a[393]||(a[393]=t("a",{id:"IncompressibleNavierStokes.RKMethods.RIA3",href:"#IncompressibleNavierStokes.RKMethods.RIA3"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.RIA3")],-1)),a[394]||(a[394]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[395]||(a[395]=e(`
    julia
    RIA3(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    RIA2.
    source
    `,3))]),t("details",p4,[t("summary",null,[a[393]||(a[393]=t("a",{id:"IncompressibleNavierStokes.RKMethods.RIA3",href:"#IncompressibleNavierStokes.RKMethods.RIA3"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.RIA3")],-1)),a[394]||(a[394]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[395]||(a[395]=e(`
    julia
    RIA3(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    RIA3.
    source
    `,3))]),t("details",h4,[t("summary",null,[a[396]||(a[396]=t("a",{id:"IncompressibleNavierStokes.RKMethods.RIIA1",href:"#IncompressibleNavierStokes.RKMethods.RIIA1"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.RIIA1")],-1)),a[397]||(a[397]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[398]||(a[398]=e(`
    julia
    RIIA1(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    RIA3.
    source
    `,3))]),t("details",h4,[t("summary",null,[a[396]||(a[396]=t("a",{id:"IncompressibleNavierStokes.RKMethods.RIIA1",href:"#IncompressibleNavierStokes.RKMethods.RIIA1"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.RIIA1")],-1)),a[397]||(a[397]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[398]||(a[398]=e(`
    julia
    RIIA1(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    RIIA1.
    source
    `,3))]),t("details",g4,[t("summary",null,[a[399]||(a[399]=t("a",{id:"IncompressibleNavierStokes.RKMethods.RIIA2",href:"#IncompressibleNavierStokes.RKMethods.RIIA2"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.RIIA2")],-1)),a[400]||(a[400]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[401]||(a[401]=e(`
    julia
    RIIA2(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    RIIA1.
    source
    `,3))]),t("details",g4,[t("summary",null,[a[399]||(a[399]=t("a",{id:"IncompressibleNavierStokes.RKMethods.RIIA2",href:"#IncompressibleNavierStokes.RKMethods.RIIA2"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.RIIA2")],-1)),a[400]||(a[400]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[401]||(a[401]=e(`
    julia
    RIIA2(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    RIIA2.
    source
    `,3))]),t("details",k4,[t("summary",null,[a[402]||(a[402]=t("a",{id:"IncompressibleNavierStokes.RKMethods.RIIA3",href:"#IncompressibleNavierStokes.RKMethods.RIIA3"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.RIIA3")],-1)),a[403]||(a[403]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[404]||(a[404]=e(`
    julia
    RIIA3(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    RIIA2.
    source
    `,3))]),t("details",k4,[t("summary",null,[a[402]||(a[402]=t("a",{id:"IncompressibleNavierStokes.RKMethods.RIIA3",href:"#IncompressibleNavierStokes.RKMethods.RIIA3"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.RIIA3")],-1)),a[403]||(a[403]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[404]||(a[404]=e(`
    julia
    RIIA3(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    RIIA3.
    source
    `,3))]),t("details",c4,[t("summary",null,[a[405]||(a[405]=t("a",{id:"IncompressibleNavierStokes.RKMethods.LIIIA2",href:"#IncompressibleNavierStokes.RKMethods.LIIIA2"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.LIIIA2")],-1)),a[406]||(a[406]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[407]||(a[407]=e(`
    julia
    LIIIA2(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    RIIA3.
    source
    `,3))]),t("details",c4,[t("summary",null,[a[405]||(a[405]=t("a",{id:"IncompressibleNavierStokes.RKMethods.LIIIA2",href:"#IncompressibleNavierStokes.RKMethods.LIIIA2"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.LIIIA2")],-1)),a[406]||(a[406]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[407]||(a[407]=e(`
    julia
    LIIIA2(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    LIIIA2.
    source
    `,3))]),t("details",u4,[t("summary",null,[a[408]||(a[408]=t("a",{id:"IncompressibleNavierStokes.RKMethods.LIIIA3",href:"#IncompressibleNavierStokes.RKMethods.LIIIA3"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.LIIIA3")],-1)),a[409]||(a[409]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[410]||(a[410]=e(`
    julia
    LIIIA3(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    LIIIA2.
    source
    `,3))]),t("details",u4,[t("summary",null,[a[408]||(a[408]=t("a",{id:"IncompressibleNavierStokes.RKMethods.LIIIA3",href:"#IncompressibleNavierStokes.RKMethods.LIIIA3"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.LIIIA3")],-1)),a[409]||(a[409]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[410]||(a[410]=e(`
    julia
    LIIIA3(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    LIIIA3.
    source
    `,3))]),a[486]||(a[486]=t("h3",{id:"Chebyshev-methods",tabindex:"-1"},[s("Chebyshev methods "),t("a",{class:"header-anchor",href:"#Chebyshev-methods","aria-label":'Permalink to "Chebyshev methods {#Chebyshev-methods}"'},"​")],-1)),t("details",H4,[t("summary",null,[a[411]||(a[411]=t("a",{id:"IncompressibleNavierStokes.RKMethods.CHDIRK3",href:"#IncompressibleNavierStokes.RKMethods.CHDIRK3"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.CHDIRK3")],-1)),a[412]||(a[412]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[413]||(a[413]=e(`
    julia
    CHDIRK3(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    LIIIA3.
    source
    `,3))]),a[486]||(a[486]=t("h3",{id:"Chebyshev-methods",tabindex:"-1"},[s("Chebyshev methods "),t("a",{class:"header-anchor",href:"#Chebyshev-methods","aria-label":'Permalink to "Chebyshev methods {#Chebyshev-methods}"'},"​")],-1)),t("details",H4,[t("summary",null,[a[411]||(a[411]=t("a",{id:"IncompressibleNavierStokes.RKMethods.CHDIRK3",href:"#IncompressibleNavierStokes.RKMethods.CHDIRK3"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.CHDIRK3")],-1)),a[412]||(a[412]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[413]||(a[413]=e(`
    julia
    CHDIRK3(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    Chebyshev based DIRK (not algebraically stable).
    source
    `,3))]),t("details",y4,[t("summary",null,[a[414]||(a[414]=t("a",{id:"IncompressibleNavierStokes.RKMethods.CHCONS3",href:"#IncompressibleNavierStokes.RKMethods.CHCONS3"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.CHCONS3")],-1)),a[415]||(a[415]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[416]||(a[416]=e(`
    julia
    CHCONS3(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    Chebyshev based DIRK (not algebraically stable).
    source
    `,3))]),t("details",y4,[t("summary",null,[a[414]||(a[414]=t("a",{id:"IncompressibleNavierStokes.RKMethods.CHCONS3",href:"#IncompressibleNavierStokes.RKMethods.CHCONS3"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.CHCONS3")],-1)),a[415]||(a[415]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[416]||(a[416]=e(`
    julia
    CHCONS3(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    CHCONS3.
    source
    `,3))]),t("details",w4,[t("summary",null,[a[417]||(a[417]=t("a",{id:"IncompressibleNavierStokes.RKMethods.CHC3",href:"#IncompressibleNavierStokes.RKMethods.CHC3"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.CHC3")],-1)),a[418]||(a[418]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[419]||(a[419]=e(`
    julia
    CHC3(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    CHCONS3.
    source
    `,3))]),t("details",w4,[t("summary",null,[a[417]||(a[417]=t("a",{id:"IncompressibleNavierStokes.RKMethods.CHC3",href:"#IncompressibleNavierStokes.RKMethods.CHC3"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.CHC3")],-1)),a[418]||(a[418]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[419]||(a[419]=e(`
    julia
    CHC3(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    Chebyshev quadrature and C(3) satisfied. Note this equals Lobatto IIIA.
    source
    `,3))]),t("details",x4,[t("summary",null,[a[420]||(a[420]=t("a",{id:"IncompressibleNavierStokes.RKMethods.CHC5",href:"#IncompressibleNavierStokes.RKMethods.CHC5"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.CHC5")],-1)),a[421]||(a[421]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[422]||(a[422]=e(`
    julia
    CHC5(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    Chebyshev quadrature and C(3) satisfied. Note this equals Lobatto IIIA.
    source
    `,3))]),t("details",x4,[t("summary",null,[a[420]||(a[420]=t("a",{id:"IncompressibleNavierStokes.RKMethods.CHC5",href:"#IncompressibleNavierStokes.RKMethods.CHC5"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.CHC5")],-1)),a[421]||(a[421]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[422]||(a[422]=e(`
    julia
    CHC5(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    CHC5.
    source
    `,3))]),a[487]||(a[487]=t("h3",{id:"Miscellaneous-Methods",tabindex:"-1"},[s("Miscellaneous Methods "),t("a",{class:"header-anchor",href:"#Miscellaneous-Methods","aria-label":'Permalink to "Miscellaneous Methods {#Miscellaneous-Methods}"'},"​")],-1)),t("details",f4,[t("summary",null,[a[423]||(a[423]=t("a",{id:"IncompressibleNavierStokes.RKMethods.Mid22",href:"#IncompressibleNavierStokes.RKMethods.Mid22"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.Mid22")],-1)),a[424]||(a[424]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[425]||(a[425]=e(`
    julia
    Mid22(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    CHC5.
    source
    `,3))]),a[487]||(a[487]=t("h3",{id:"Miscellaneous-Methods",tabindex:"-1"},[s("Miscellaneous Methods "),t("a",{class:"header-anchor",href:"#Miscellaneous-Methods","aria-label":'Permalink to "Miscellaneous Methods {#Miscellaneous-Methods}"'},"​")],-1)),t("details",f4,[t("summary",null,[a[423]||(a[423]=t("a",{id:"IncompressibleNavierStokes.RKMethods.Mid22",href:"#IncompressibleNavierStokes.RKMethods.Mid22"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.Mid22")],-1)),a[424]||(a[424]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[425]||(a[425]=e(`
    julia
    Mid22(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    Midpoint 22 method.
    source
    `,3))]),t("details",b4,[t("summary",null,[a[426]||(a[426]=t("a",{id:"IncompressibleNavierStokes.RKMethods.MTE22",href:"#IncompressibleNavierStokes.RKMethods.MTE22"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.MTE22")],-1)),a[427]||(a[427]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[428]||(a[428]=e(`
    julia
    MTE22(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    Midpoint 22 method.
    source
    `,3))]),t("details",b4,[t("summary",null,[a[426]||(a[426]=t("a",{id:"IncompressibleNavierStokes.RKMethods.MTE22",href:"#IncompressibleNavierStokes.RKMethods.MTE22"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.MTE22")],-1)),a[427]||(a[427]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[428]||(a[428]=e(`
    julia
    MTE22(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    Minimal truncation error 22 method (Heun).
    source
    `,3))]),t("details",M4,[t("summary",null,[a[429]||(a[429]=t("a",{id:"IncompressibleNavierStokes.RKMethods.CN22",href:"#IncompressibleNavierStokes.RKMethods.CN22"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.CN22")],-1)),a[430]||(a[430]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[431]||(a[431]=e(`
    julia
    CN22(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    Minimal truncation error 22 method (Heun).
    source
    `,3))]),t("details",M4,[t("summary",null,[a[429]||(a[429]=t("a",{id:"IncompressibleNavierStokes.RKMethods.CN22",href:"#IncompressibleNavierStokes.RKMethods.CN22"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.CN22")],-1)),a[430]||(a[430]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[431]||(a[431]=e(`
    julia
    CN22(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    Crank-Nicholson.
    source
    `,3))]),t("details",L4,[t("summary",null,[a[432]||(a[432]=t("a",{id:"IncompressibleNavierStokes.RKMethods.Heun33",href:"#IncompressibleNavierStokes.RKMethods.Heun33"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.Heun33")],-1)),a[433]||(a[433]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[434]||(a[434]=e(`
    julia
    Heun33(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    Crank-Nicholson.
    source
    `,3))]),t("details",L4,[t("summary",null,[a[432]||(a[432]=t("a",{id:"IncompressibleNavierStokes.RKMethods.Heun33",href:"#IncompressibleNavierStokes.RKMethods.Heun33"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.Heun33")],-1)),a[433]||(a[433]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[434]||(a[434]=e(`
    julia
    Heun33(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    Heun33.
    source
    `,3))]),t("details",v4,[t("summary",null,[a[435]||(a[435]=t("a",{id:"IncompressibleNavierStokes.RKMethods.RK33C2",href:"#IncompressibleNavierStokes.RKMethods.RK33C2"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.RK33C2")],-1)),a[436]||(a[436]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[437]||(a[437]=e(`
    julia
    RK33C2(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    Heun33.
    source
    `,3))]),t("details",v4,[t("summary",null,[a[435]||(a[435]=t("a",{id:"IncompressibleNavierStokes.RKMethods.RK33C2",href:"#IncompressibleNavierStokes.RKMethods.RK33C2"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.RK33C2")],-1)),a[436]||(a[436]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[437]||(a[437]=e(`
    julia
    RK33C2(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    RK3 satisfying C(2) for i=3.
    source
    `,3))]),t("details",E4,[t("summary",null,[a[438]||(a[438]=t("a",{id:"IncompressibleNavierStokes.RKMethods.RK33P2",href:"#IncompressibleNavierStokes.RKMethods.RK33P2"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.RK33P2")],-1)),a[439]||(a[439]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[440]||(a[440]=e(`
    julia
    RK33P2(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    RK3 satisfying C(2) for i=3.
    source
    `,3))]),t("details",E4,[t("summary",null,[a[438]||(a[438]=t("a",{id:"IncompressibleNavierStokes.RKMethods.RK33P2",href:"#IncompressibleNavierStokes.RKMethods.RK33P2"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.RK33P2")],-1)),a[439]||(a[439]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[440]||(a[440]=e(`
    julia
    RK33P2(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    RK3 satisfying the second order condition for the pressure.
    source
    `,3))]),t("details",Z4,[t("summary",null,[a[441]||(a[441]=t("a",{id:"IncompressibleNavierStokes.RKMethods.RK44",href:"#IncompressibleNavierStokes.RKMethods.RK44"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.RK44")],-1)),a[442]||(a[442]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[443]||(a[443]=e(`
    julia
    RK44(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    RK3 satisfying the second order condition for the pressure.
    source
    `,3))]),t("details",Z4,[t("summary",null,[a[441]||(a[441]=t("a",{id:"IncompressibleNavierStokes.RKMethods.RK44",href:"#IncompressibleNavierStokes.RKMethods.RK44"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.RK44")],-1)),a[442]||(a[442]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[443]||(a[443]=e(`
    julia
    RK44(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    Classical fourth order.
    source
    `,3))]),t("details",j4,[t("summary",null,[a[444]||(a[444]=t("a",{id:"IncompressibleNavierStokes.RKMethods.RK44C2",href:"#IncompressibleNavierStokes.RKMethods.RK44C2"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.RK44C2")],-1)),a[445]||(a[445]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[446]||(a[446]=e(`
    julia
    RK44C2(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    Classical fourth order.
    source
    `,3))]),t("details",j4,[t("summary",null,[a[444]||(a[444]=t("a",{id:"IncompressibleNavierStokes.RKMethods.RK44C2",href:"#IncompressibleNavierStokes.RKMethods.RK44C2"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.RK44C2")],-1)),a[445]||(a[445]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[446]||(a[446]=e(`
    julia
    RK44C2(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    RK4 satisfying C(2) for i=3.
    source
    `,3))]),t("details",D4,[t("summary",null,[a[447]||(a[447]=t("a",{id:"IncompressibleNavierStokes.RKMethods.RK44C23",href:"#IncompressibleNavierStokes.RKMethods.RK44C23"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.RK44C23")],-1)),a[448]||(a[448]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[449]||(a[449]=e(`
    julia
    RK44C23(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    RK4 satisfying C(2) for i=3.
    source
    `,3))]),t("details",D4,[t("summary",null,[a[447]||(a[447]=t("a",{id:"IncompressibleNavierStokes.RKMethods.RK44C23",href:"#IncompressibleNavierStokes.RKMethods.RK44C23"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.RK44C23")],-1)),a[448]||(a[448]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[449]||(a[449]=e(`
    julia
    RK44C23(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    RK4 satisfying C(2) for i=3 and c2=c3.
    source
    `,3))]),t("details",S4,[t("summary",null,[a[450]||(a[450]=t("a",{id:"IncompressibleNavierStokes.RKMethods.RK44P2",href:"#IncompressibleNavierStokes.RKMethods.RK44P2"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.RK44P2")],-1)),a[451]||(a[451]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[452]||(a[452]=e(`
    julia
    RK44P2(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    RK4 satisfying C(2) for i=3 and c2=c3.
    source
    `,3))]),t("details",S4,[t("summary",null,[a[450]||(a[450]=t("a",{id:"IncompressibleNavierStokes.RKMethods.RK44P2",href:"#IncompressibleNavierStokes.RKMethods.RK44P2"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.RK44P2")],-1)),a[451]||(a[451]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[452]||(a[452]=e(`
    julia
    RK44P2(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    RK4 satisfying the second order condition for the pressure (but not third order).
    source
    `,3))]),a[488]||(a[488]=t("h3",{id:"DSRK-Methods",tabindex:"-1"},[s("DSRK Methods "),t("a",{class:"header-anchor",href:"#DSRK-Methods","aria-label":'Permalink to "DSRK Methods {#DSRK-Methods}"'},"​")],-1)),t("details",V4,[t("summary",null,[a[453]||(a[453]=t("a",{id:"IncompressibleNavierStokes.RKMethods.DSso2",href:"#IncompressibleNavierStokes.RKMethods.DSso2"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.DSso2")],-1)),a[454]||(a[454]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[455]||(a[455]=e(`
    julia
    DSso2(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    RK4 satisfying the second order condition for the pressure (but not third order).
    source
    `,3))]),a[488]||(a[488]=t("h3",{id:"DSRK-Methods",tabindex:"-1"},[s("DSRK Methods "),t("a",{class:"header-anchor",href:"#DSRK-Methods","aria-label":'Permalink to "DSRK Methods {#DSRK-Methods}"'},"​")],-1)),t("details",V4,[t("summary",null,[a[453]||(a[453]=t("a",{id:"IncompressibleNavierStokes.RKMethods.DSso2",href:"#IncompressibleNavierStokes.RKMethods.DSso2"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.DSso2")],-1)),a[454]||(a[454]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[455]||(a[455]=e(`
    julia
    DSso2(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    CBM's DSRKso2.
    source
    `,3))]),t("details",I4,[t("summary",null,[a[456]||(a[456]=t("a",{id:"IncompressibleNavierStokes.RKMethods.DSRK2",href:"#IncompressibleNavierStokes.RKMethods.DSRK2"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.DSRK2")],-1)),a[457]||(a[457]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[458]||(a[458]=e(`
    julia
    DSRK2(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    CBM's DSRKso2.
    source
    `,3))]),t("details",I4,[t("summary",null,[a[456]||(a[456]=t("a",{id:"IncompressibleNavierStokes.RKMethods.DSRK2",href:"#IncompressibleNavierStokes.RKMethods.DSRK2"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.DSRK2")],-1)),a[457]||(a[457]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[458]||(a[458]=e(`
    julia
    DSRK2(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    CBM's DSRK2.
    source
    `,3))]),t("details",F4,[t("summary",null,[a[459]||(a[459]=t("a",{id:"IncompressibleNavierStokes.RKMethods.DSRK3",href:"#IncompressibleNavierStokes.RKMethods.DSRK3"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.DSRK3")],-1)),a[460]||(a[460]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[461]||(a[461]=e(`
    julia
    DSRK3(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    CBM's DSRK2.
    source
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    julia
    DSRK3(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    Zennaro's DSRK3.
    source
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    julia
    NSSP21(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    Zennaro's DSRK3.
    source
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    NSSP21(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    NSSP21.
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    julia
    NSSP32(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    NSSP21.
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    julia
    NSSP32(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    NSSP32.
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    julia
    NSSP33(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    NSSP32.
    source
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    julia
    NSSP33(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
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    julia
    NSSP53(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    NSSP33.
    source
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    NSSP53(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    NSSP53.
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    NSSP53.
    source
    `,3))])])}const X4=n(r,[["render",A4]]);export{U4 as __pageData,X4 as default};
diff --git a/dev/assets/manual_time.md.3eTgRN26.lean.js b/dev/assets/manual_time.md.BJBCC_BN.lean.js
similarity index 99%
rename from dev/assets/manual_time.md.3eTgRN26.lean.js
rename to dev/assets/manual_time.md.BJBCC_BN.lean.js
index d664e51e2..85c06a11c 100644
--- a/dev/assets/manual_time.md.3eTgRN26.lean.js
+++ b/dev/assets/manual_time.md.BJBCC_BN.lean.js
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This term ensures that the time-continuous discrete velocity field ")),t("mjx-container",c,[(l(),Q("svg",u,a[6]||(a[6]=[e('',1)]))),a[7]||(a[7]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"u"),t("mo",{stretchy:"false"},"("),t("mi",null,"t"),t("mo",{stretchy:"false"},")")])],-1))]),a[10]||(a[10]=s(" stays divergence free (conserves mass). However, if we directly discretize this system in time, the mass preservation may actually not be respected. For this, we will change the definition of the pressure such that the time-discretized velocity field is divergence free at each time step and each time sub-step (to be defined in the following)."))]),t("p",null,[a[41]||(a[41]=s("Consider the interval ")),t("mjx-container",H,[(l(),Q("svg",y,a[11]||(a[11]=[e('',1)]))),a[12]||(a[12]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mo",{stretchy:"false"},"["),t("mn",null,"0"),t("mo",null,","),t("mi",null,"T"),t("mo",{stretchy:"false"},"]")])],-1))]),a[42]||(a[42]=s(" for some simulation time ")),t("mjx-container",w,[(l(),Q("svg",x,a[13]||(a[13]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D447",d:"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z",style:{"stroke-width":"3"}})])])],-1)]))),a[14]||(a[14]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"T")])],-1))]),a[43]||(a[43]=s(". We will divide it into ")),t("mjx-container",f,[(l(),Q("svg",b,a[15]||(a[15]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D441",d:"M234 637Q231 637 226 637Q201 637 196 638T191 649Q191 676 202 682Q204 683 299 683Q376 683 387 683T401 677Q612 181 616 168L670 381Q723 592 723 606Q723 633 659 637Q635 637 635 648Q635 650 637 660Q641 676 643 679T653 683Q656 683 684 682T767 680Q817 680 843 681T873 682Q888 682 888 672Q888 650 880 642Q878 637 858 637Q787 633 769 597L620 7Q618 0 599 0Q585 0 582 2Q579 5 453 305L326 604L261 344Q196 88 196 79Q201 46 268 46H278Q284 41 284 38T282 19Q278 6 272 0H259Q228 2 151 2Q123 2 100 2T63 2T46 1Q31 1 31 10Q31 14 34 26T39 40Q41 46 62 46Q130 49 150 85Q154 91 221 362L289 634Q287 635 234 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")),t("mjx-container",D,[(l(),Q("svg",S,a[23]||(a[23]=[e('',1)]))),a[24]||(a[24]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msup",null,[t("mi",null,"t"),t("mi",null,"N")]),t("mo",null,"="),t("mi",null,"T")])],-1))]),a[48]||(a[48]=s(", and increment ")),t("mjx-container",V,[(l(),Q("svg",I,a[25]||(a[25]=[e('',1)]))),a[26]||(a[26]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 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We define ")),t("mjx-container",F,[(l(),Q("svg",R,a[27]||(a[27]=[e('',1)]))),a[28]||(a[28]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msup",null,[t("mi",null,"u"),t("mi",null,"n")]),t("mo",null,"≈"),t("mi",null,"u"),t("mo",{stretchy:"false"},"("),t("msup",null,[t("mi",null,"t"),t("mi",null,"n")]),t("mo",{stretchy:"false"},")")])],-1))]),a[50]||(a[50]=s(" as an approximation to the exact discrete velocity field ")),t("mjx-container",C,[(l(),Q("svg",N,a[29]||(a[29]=[e('',1)]))),a[30]||(a[30]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"u"),t("mo",{stretchy:"false"},"("),t("msup",null,[t("mi",null,"t"),t("mi",null,"n")]),t("mo",{stretchy:"false"},")")])],-1))]),a[51]||(a[51]=s(", with ")),t("mjx-container",K,[(l(),Q("svg",A,a[31]||(a[31]=[e('',1)]))),a[32]||(a[32]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msup",null,[t("mi",null,"u"),t("mn",null,"0")]),t("mo",null,"="),t("mi",null,"u"),t("mo",{stretchy:"false"},"("),t("mn",null,"0"),t("mo",{stretchy:"false"},")")])],-1))]),a[52]||(a[52]=s(" starting from the exact initial conditions. We say that the time integration scheme (definition of ")),t("mjx-container",B,[(l(),Q("svg",O,a[33]||(a[33]=[e('',1)]))),a[34]||(a[34]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msup",null,[t("mi",null,"u"),t("mi",null,"n")])])],-1))]),a[53]||(a[53]=s(") is accurate to the order ")),t("mjx-container",P,[(l(),Q("svg",G,a[35]||(a[35]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D45F",d:"M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 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")),t("mjx-container",z,[(l(),Q("svg",J,a[37]||(a[37]=[e('',1)]))),a[38]||(a[38]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 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    julia
    abstract type AbstractODEMethod{T}
    Abstract ODE method.
    Fields
    source
    ',4))]),t("details",q,[t("summary",null,[a[60]||(a[60]=t("a",{id:"IncompressibleNavierStokes.ode_method_cache",href:"#IncompressibleNavierStokes.ode_method_cache"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.ode_method_cache")],-1)),a[61]||(a[61]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[62]||(a[62]=e('
    julia
    ode_method_cache(method, setup, u, temp)
    Get time stepper cache for the given ODE method.
    source
    ',3))]),t("details",$,[t("summary",null,[a[63]||(a[63]=t("a",{id:"IncompressibleNavierStokes.runge_kutta_method",href:"#IncompressibleNavierStokes.runge_kutta_method"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.runge_kutta_method")],-1)),a[64]||(a[64]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[65]||(a[65]=e(`
    julia
    runge_kutta_method(
     A,
     b,
     c,
     r;
     T,
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod, IncompressibleNavierStokes.ImplicitRungeKuttaMethod}
    Get Runge Kutta method. The function checks whether the method is explicit.
    p_add_solve: whether to add a pressure solve step to the method.
    For implicit RK methods: newton_type, maxiter, abstol, reltol.
    source
    `,5))]),t("details",Y,[t("summary",null,[a[66]||(a[66]=t("a",{id:"IncompressibleNavierStokes.create_stepper",href:"#IncompressibleNavierStokes.create_stepper"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.create_stepper")],-1)),a[67]||(a[67]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[68]||(a[68]=e('
    julia
    create_stepper(method; setup, psolver, u, temp, t, n)
    Create time stepper.
    source
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    julia
    timestep(method, stepper, Δt; θ = nothing)
    Perform one time step.
    Non-mutating/allocating/out-of-place version.
    See also timestep!.
    source
    ',5))]),t("details",t1,[t("summary",null,[a[72]||(a[72]=t("a",{id:"IncompressibleNavierStokes.timestep!",href:"#IncompressibleNavierStokes.timestep!"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.timestep!")],-1)),a[73]||(a[73]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[74]||(a[74]=e('
    julia
    timestep!(method, stepper, Δt; θ = nothing, cache)
    Perform one time step>
    Mutating/non-allocating/in-place version.
    See also timestep.
    source
    ',5))]),a[479]||(a[479]=t("h2",{id:"Adams-Bashforth-Crank-Nicolson-method",tabindex:"-1"},[s("Adams-Bashforth Crank-Nicolson method "),t("a",{class:"header-anchor",href:"#Adams-Bashforth-Crank-Nicolson-method","aria-label":'Permalink to "Adams-Bashforth Crank-Nicolson method {#Adams-Bashforth-Crank-Nicolson-method}"'},"​")],-1)),t("details",a1,[t("summary",null,[a[75]||(a[75]=t("a",{id:"IncompressibleNavierStokes.AdamsBashforthCrankNicolsonMethod",href:"#IncompressibleNavierStokes.AdamsBashforthCrankNicolsonMethod"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.AdamsBashforthCrankNicolsonMethod")],-1)),a[76]||(a[76]=s()),i(T,{type:"info",class:"jlObjectType jlType",text:"Type"})]),a[151]||(a[151]=e('
    julia
    struct AdamsBashforthCrankNicolsonMethod{T, M} <: IncompressibleNavierStokes.AbstractODEMethod{T}
    IMEX AB-CN: Adams-Bashforth for explicit convection (parameters α₁ and α₂) and Crank-Nicolson for implicit diffusion (implicitness θ). The method is second order for θ = 1/2.
    The LU decomposition of the LHS matrix is computed every time the time step changes.
    Note that, in contrast to explicit methods, the pressure from previous time steps has an influence on the accuracy of the velocity.
    ',4)),t("p",null,[a[91]||(a[91]=s("We here require that the time step ")),t("mjx-container",s1,[(l(),Q("svg",e1,a[77]||(a[77]=[e('',1)]))),a[78]||(a[78]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",{mathvariant:"normal"},"Δ"),t("mi",null,"t")])],-1))]),a[92]||(a[92]=s(" is constant. This methods uses Adams-Bashforth for the convective terms and Crank-Nicolson stepping for the diffusion and body force terms. Given the velocity field ")),t("mjx-container",Q1,[(l(),Q("svg",l1,a[79]||(a[79]=[e('',1)]))),a[80]||(a[80]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"u"),t("mn",null,"0")]),t("mo",null,"="),t("mi",null,"u"),t("mo",{stretchy:"false"},"("),t("msub",null,[t("mi",null,"t"),t("mn",null,"0")]),t("mo",{stretchy:"false"},")")])],-1))]),a[93]||(a[93]=s(" at a time ")),t("mjx-container",T1,[(l(),Q("svg",i1,a[81]||(a[81]=[e('',1)]))),a[82]||(a[82]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 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")),t("mjx-container",u1,[(l(),Q("svg",H1,a[101]||(a[101]=[e('',1)]))),a[102]||(a[102]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"θ"),t("mo",null,"∈"),t("mo",{stretchy:"false"},"["),t("mn",null,"0"),t("mo",null,","),t("mn",null,"1"),t("mo",{stretchy:"false"},"]")])],-1))]),a[112]||(a[112]=s(" is the Crank-Nicolson parameter (")),t("mjx-container",y1,[(l(),Q("svg",w1,a[103]||(a[103]=[e('',1)]))),a[104]||(a[104]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 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a tentative velocity yet to be made divergence free. We can group the terms containing ")),t("mjx-container",L1,[(l(),Q("svg",v1,a[109]||(a[109]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D463",d:"M173 380Q173 405 154 405Q130 405 104 376T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Q21 294 29 316T53 368T97 419T160 441Q202 441 225 417T249 361Q249 344 246 335Q246 329 231 291T200 202T182 113Q182 86 187 69Q200 26 250 26Q287 26 319 60T369 139T398 222T409 277Q409 300 401 317T383 343T365 361T357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159T347 40T241 -11Q177 -11 139 22Q102 54 102 117Q102 148 110 181T151 298Q173 362 173 380Z",style:{"stroke-width":"3"}})])])],-1)]))),a[110]||(a[110]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 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Assuming ")),t("mjx-container",I1,[(l(),Q("svg",F1,a[123]||(a[123]=[e('',1)]))),a[124]||(a[124]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",{mathvariant:"normal"},"Δ"),t("mi",null,"t")])],-1))]),a[128]||(a[128]=s(" is constant, we can precompute a Cholesky factorization of this matrix before starting time stepping."))]),t("p",null,[a[131]||(a[131]=s("We then compute the pressure difference ")),t("mjx-container",R1,[(l(),Q("svg",C1,a[129]||(a[129]=[e('',1)]))),a[130]||(a[130]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 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")),t("mjx-container",G1,[(l(),Q("svg",z1,a[141]||(a[141]=[e('',1)]))),a[142]||(a[142]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"p"),t("mo",null,"="),t("msub",null,[t("mi",null,"p"),t("mn",null,"0")]),t("mo",null,"+"),t("mi",{mathvariant:"normal"},"Δ"),t("mi",null,"p")])],-1))]),a[148]||(a[148]=s(". However, since this pressure is reused in the next time step, we perform an additional pressure solve to avoid accumulating first order errors. The resulting pressure ")),t("mjx-container",J1,[(l(),Q("svg",U1,a[143]||(a[143]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D45D",d:"M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z",style:{"stroke-width":"3"}})])])],-1)]))),a[144]||(a[144]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"p")])],-1))]),a[149]||(a[149]=s(" is then accurate to the same order as ")),t("mjx-container",X1,[(l(),Q("svg",W1,a[145]||(a[145]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D462",d:"M21 287Q21 295 30 318T55 370T99 420T158 442Q204 442 227 417T250 358Q250 340 216 246T182 105Q182 62 196 45T238 27T291 44T328 78L339 95Q341 99 377 247Q407 367 413 387T427 416Q444 431 463 431Q480 431 488 421T496 402L420 84Q419 79 419 68Q419 43 426 35T447 26Q469 29 482 57T512 145Q514 153 532 153Q551 153 551 144Q550 139 549 130T540 98T523 55T498 17T462 -8Q454 -10 438 -10Q372 -10 347 46Q345 45 336 36T318 21T296 6T267 -6T233 -11Q189 -11 155 7Q103 38 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z",style:{"stroke-width":"3"}})])])],-1)]))),a[146]||(a[146]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"u")])],-1))]),a[150]||(a[150]=s("."))]),a[152]||(a[152]=e('
    Fields
    • α₁
    • α₂
    • θ
    • p_add_solve
    • method_startup
    source
    ',3))]),a[480]||(a[480]=t("h2",{id:"One-leg-beta-method",tabindex:"-1"},[s("One-leg beta method "),t("a",{class:"header-anchor",href:"#One-leg-beta-method","aria-label":'Permalink to "One-leg beta method {#One-leg-beta-method}"'},"​")],-1)),t("details",q1,[t("summary",null,[a[153]||(a[153]=t("a",{id:"IncompressibleNavierStokes.OneLegMethod",href:"#IncompressibleNavierStokes.OneLegMethod"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.OneLegMethod")],-1)),a[154]||(a[154]=s()),i(T,{type:"info",class:"jlObjectType jlType",text:"Type"})]),a[202]||(a[202]=e('
    julia
    struct OneLegMethod{T, M} <: IncompressibleNavierStokes.AbstractODEMethod{T}
    Explicit one-leg β-method following symmetry-preserving discretization of turbulent flow. See Verstappen and Veldman [14] [16] for details.
    ',2)),t("p",null,[a[175]||(a[175]=s("We here require that the time step ")),t("mjx-container",$1,[(l(),Q("svg",Y1,a[155]||(a[155]=[e('',1)]))),a[156]||(a[156]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",{mathvariant:"normal"},"Δ"),t("mi",null,"t")])],-1))]),a[176]||(a[176]=s(" is constant. Given the velocity ")),t("mjx-container",_1,[(l(),Q("svg",t2,a[157]||(a[157]=[e('',1)]))),a[158]||(a[158]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"u"),t("mn",null,"0")])])],-1))]),a[177]||(a[177]=s(" and pressure ")),t("mjx-container",a2,[(l(),Q("svg",s2,a[159]||(a[159]=[e('',1)]))),a[160]||(a[160]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 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    julia
    abstract type AbstractRungeKuttaMethod{T} <: IncompressibleNavierStokes.AbstractODEMethod{T}
    Abstract Runge Kutta method.
    Fields
    source
    ',4))]),t("details",j2,[t("summary",null,[a[211]||(a[211]=t("a",{id:"IncompressibleNavierStokes.ExplicitRungeKuttaMethod",href:"#IncompressibleNavierStokes.ExplicitRungeKuttaMethod"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.ExplicitRungeKuttaMethod")],-1)),a[212]||(a[212]=s()),i(T,{type:"info",class:"jlObjectType jlType",text:"Type"})]),a[313]||(a[313]=e('
    julia
    struct ExplicitRungeKuttaMethod{T} <: IncompressibleNavierStokes.AbstractRungeKuttaMethod{T}
    Explicit Runge Kutta method. See Sanderse [17].
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The space-discrete divergence-freeness is thus perfectly preserved, even though the time discretization introduces other errors."))]),a[314]||(a[314]=e('
    Fields
    • A
    • b
    • c
    • r
    • p_add_solve
    source
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    julia
    struct ImplicitRungeKuttaMethod{T} <: IncompressibleNavierStokes.AbstractRungeKuttaMethod{T}
    Implicit Runge Kutta method. See Sanderse [18].
    The implicit linear system is solved at each time step using Newton's method. The newton_type may be one of the following:
    • :no: Replace iteration matrix with I/Δt (no Jacobian)
    • :approximate: Build Jacobian once before iterations only
    • :full: Build Jacobian at each iteration
    Fields
    • A
    • b
    • c
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    • p_add_solve
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    Set up Butcher arrays A, b, and c, as well as and SSP coefficient r. For families of methods, optional input s is the number of stages.
    Original (MATLAB) by David Ketcheson, extended by Benjamin Sanderse.
    Exports
    source
    ',5))]),t("details",P3,[t("summary",null,[a[321]||(a[321]=t("a",{id:"IncompressibleNavierStokes.LMWray3",href:"#IncompressibleNavierStokes.LMWray3"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.LMWray3")],-1)),a[322]||(a[322]=s()),i(T,{type:"info",class:"jlObjectType jlType",text:"Type"})]),a[323]||(a[323]=e('
    julia
    struct LMWray3{T} <: IncompressibleNavierStokes.AbstractRungeKuttaMethod{T}
    Low memory Wray 3rd order scheme. Uses 3 vector fields and one scalar field.
    Fields
    source
    ',4))]),a[482]||(a[482]=t("h3",{id:"Explicit-Methods",tabindex:"-1"},[s("Explicit Methods "),t("a",{class:"header-anchor",href:"#Explicit-Methods","aria-label":'Permalink to "Explicit Methods {#Explicit-Methods}"'},"​")],-1)),t("details",G3,[t("summary",null,[a[324]||(a[324]=t("a",{id:"IncompressibleNavierStokes.RKMethods.FE11",href:"#IncompressibleNavierStokes.RKMethods.FE11"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.FE11")],-1)),a[325]||(a[325]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[326]||(a[326]=e(`
    julia
    FE11(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod, IncompressibleNavierStokes.ImplicitRungeKuttaMethod}
    Get Runge Kutta method. The function checks whether the method is explicit.
    p_add_solve: whether to add a pressure solve step to the method.
    For implicit RK methods: newton_type, maxiter, abstol, reltol.
    source
    `,5))]),t("details",Y,[t("summary",null,[a[66]||(a[66]=t("a",{id:"IncompressibleNavierStokes.create_stepper",href:"#IncompressibleNavierStokes.create_stepper"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.create_stepper")],-1)),a[67]||(a[67]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[68]||(a[68]=e('
    julia
    create_stepper(method; setup, psolver, u, temp, t, n)
    Create time stepper.
    source
    ',3))]),t("details",_,[t("summary",null,[a[69]||(a[69]=t("a",{id:"IncompressibleNavierStokes.timestep",href:"#IncompressibleNavierStokes.timestep"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.timestep")],-1)),a[70]||(a[70]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[71]||(a[71]=e('
    julia
    timestep(method, stepper, Δt; θ = nothing)
    Perform one time step.
    Non-mutating/allocating/out-of-place version.
    See also timestep!.
    source
    ',5))]),t("details",t1,[t("summary",null,[a[72]||(a[72]=t("a",{id:"IncompressibleNavierStokes.timestep!",href:"#IncompressibleNavierStokes.timestep!"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.timestep!")],-1)),a[73]||(a[73]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[74]||(a[74]=e('
    julia
    timestep!(method, stepper, Δt; θ = nothing, cache)
    Perform one time step>
    Mutating/non-allocating/in-place version.
    See also timestep.
    source
    ',5))]),a[479]||(a[479]=t("h2",{id:"Adams-Bashforth-Crank-Nicolson-method",tabindex:"-1"},[s("Adams-Bashforth Crank-Nicolson method "),t("a",{class:"header-anchor",href:"#Adams-Bashforth-Crank-Nicolson-method","aria-label":'Permalink to "Adams-Bashforth Crank-Nicolson method {#Adams-Bashforth-Crank-Nicolson-method}"'},"​")],-1)),t("details",a1,[t("summary",null,[a[75]||(a[75]=t("a",{id:"IncompressibleNavierStokes.AdamsBashforthCrankNicolsonMethod",href:"#IncompressibleNavierStokes.AdamsBashforthCrankNicolsonMethod"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.AdamsBashforthCrankNicolsonMethod")],-1)),a[76]||(a[76]=s()),i(T,{type:"info",class:"jlObjectType jlType",text:"Type"})]),a[151]||(a[151]=e('
    julia
    struct AdamsBashforthCrankNicolsonMethod{T, M} <: IncompressibleNavierStokes.AbstractODEMethod{T}
    IMEX AB-CN: Adams-Bashforth for explicit convection (parameters α₁ and α₂) and Crank-Nicolson for implicit diffusion (implicitness θ). The method is second order for θ = 1/2.
    The LU decomposition of the LHS matrix is computed every time the time step changes.
    Note that, in contrast to explicit methods, the pressure from previous time steps has an influence on the accuracy of the velocity.
    ',4)),t("p",null,[a[91]||(a[91]=s("We here require that the time step ")),t("mjx-container",s1,[(l(),Q("svg",e1,a[77]||(a[77]=[e('',1)]))),a[78]||(a[78]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",{mathvariant:"normal"},"Δ"),t("mi",null,"t")])],-1))]),a[92]||(a[92]=s(" is constant. This methods uses Adams-Bashforth for the convective terms and Crank-Nicolson stepping for the diffusion and body force terms. Given the velocity field ")),t("mjx-container",Q1,[(l(),Q("svg",l1,a[79]||(a[79]=[e('',1)]))),a[80]||(a[80]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"u"),t("mn",null,"0")]),t("mo",null,"="),t("mi",null,"u"),t("mo",{stretchy:"false"},"("),t("msub",null,[t("mi",null,"t"),t("mn",null,"0")]),t("mo",{stretchy:"false"},")")])],-1))]),a[93]||(a[93]=s(" at a time ")),t("mjx-container",T1,[(l(),Q("svg",i1,a[81]||(a[81]=[e('',1)]))),a[82]||(a[82]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 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")),t("mjx-container",h1,[(l(),Q("svg",g1,a[89]||(a[89]=[e('',1)]))),a[90]||(a[90]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"t"),t("mo",null,"="),t("msub",null,[t("mi",null,"t"),t("mn",null,"0")]),t("mo",null,"+"),t("mi",{mathvariant:"normal"},"Δ"),t("mi",null,"t")])],-1))]),a[98]||(a[98]=s(" is defined by first computing a tentative velocity:"))]),t("mjx-container",k1,[(l(),Q("svg",c1,a[99]||(a[99]=[e('',1)]))),a[100]||(a[100]=t("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 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")),t("mjx-container",u1,[(l(),Q("svg",H1,a[101]||(a[101]=[e('',1)]))),a[102]||(a[102]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"θ"),t("mo",null,"∈"),t("mo",{stretchy:"false"},"["),t("mn",null,"0"),t("mo",null,","),t("mn",null,"1"),t("mo",{stretchy:"false"},"]")])],-1))]),a[112]||(a[112]=s(" is the Crank-Nicolson parameter (")),t("mjx-container",y1,[(l(),Q("svg",w1,a[103]||(a[103]=[e('',1)]))),a[104]||(a[104]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 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a tentative velocity yet to be made divergence free. We can group the terms containing ")),t("mjx-container",L1,[(l(),Q("svg",v1,a[109]||(a[109]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D463",d:"M173 380Q173 405 154 405Q130 405 104 376T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Q21 294 29 316T53 368T97 419T160 441Q202 441 225 417T249 361Q249 344 246 335Q246 329 231 291T200 202T182 113Q182 86 187 69Q200 26 250 26Q287 26 319 60T369 139T398 222T409 277Q409 300 401 317T383 343T365 361T357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159T347 40T241 -11Q177 -11 139 22Q102 54 102 117Q102 148 110 181T151 298Q173 362 173 380Z",style:{"stroke-width":"3"}})])])],-1)]))),a[110]||(a[110]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 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Assuming ")),t("mjx-container",I1,[(l(),Q("svg",F1,a[123]||(a[123]=[e('',1)]))),a[124]||(a[124]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",{mathvariant:"normal"},"Δ"),t("mi",null,"t")])],-1))]),a[128]||(a[128]=s(" is constant, we can precompute a Cholesky factorization of this matrix before starting time stepping."))]),t("p",null,[a[131]||(a[131]=s("We then compute the pressure difference ")),t("mjx-container",R1,[(l(),Q("svg",C1,a[129]||(a[129]=[e('',1)]))),a[130]||(a[130]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 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")),t("mjx-container",G1,[(l(),Q("svg",z1,a[141]||(a[141]=[e('',1)]))),a[142]||(a[142]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"p"),t("mo",null,"="),t("msub",null,[t("mi",null,"p"),t("mn",null,"0")]),t("mo",null,"+"),t("mi",{mathvariant:"normal"},"Δ"),t("mi",null,"p")])],-1))]),a[148]||(a[148]=s(". However, since this pressure is reused in the next time step, we perform an additional pressure solve to avoid accumulating first order errors. The resulting pressure ")),t("mjx-container",J1,[(l(),Q("svg",U1,a[143]||(a[143]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D45D",d:"M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z",style:{"stroke-width":"3"}})])])],-1)]))),a[144]||(a[144]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"p")])],-1))]),a[149]||(a[149]=s(" is then accurate to the same order as ")),t("mjx-container",X1,[(l(),Q("svg",W1,a[145]||(a[145]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D462",d:"M21 287Q21 295 30 318T55 370T99 420T158 442Q204 442 227 417T250 358Q250 340 216 246T182 105Q182 62 196 45T238 27T291 44T328 78L339 95Q341 99 377 247Q407 367 413 387T427 416Q444 431 463 431Q480 431 488 421T496 402L420 84Q419 79 419 68Q419 43 426 35T447 26Q469 29 482 57T512 145Q514 153 532 153Q551 153 551 144Q550 139 549 130T540 98T523 55T498 17T462 -8Q454 -10 438 -10Q372 -10 347 46Q345 45 336 36T318 21T296 6T267 -6T233 -11Q189 -11 155 7Q103 38 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z",style:{"stroke-width":"3"}})])])],-1)]))),a[146]||(a[146]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"u")])],-1))]),a[150]||(a[150]=s("."))]),a[152]||(a[152]=e('
    Fields
    • α₁
    • α₂
    • θ
    • p_add_solve
    • method_startup
    source
    ',3))]),a[480]||(a[480]=t("h2",{id:"One-leg-beta-method",tabindex:"-1"},[s("One-leg beta method "),t("a",{class:"header-anchor",href:"#One-leg-beta-method","aria-label":'Permalink to "One-leg beta method {#One-leg-beta-method}"'},"​")],-1)),t("details",q1,[t("summary",null,[a[153]||(a[153]=t("a",{id:"IncompressibleNavierStokes.OneLegMethod",href:"#IncompressibleNavierStokes.OneLegMethod"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.OneLegMethod")],-1)),a[154]||(a[154]=s()),i(T,{type:"info",class:"jlObjectType jlType",text:"Type"})]),a[202]||(a[202]=e('
    julia
    struct OneLegMethod{T, M} <: IncompressibleNavierStokes.AbstractODEMethod{T}
    Explicit one-leg β-method following symmetry-preserving discretization of turbulent flow. See Verstappen and Veldman [14] [16] for details.
    ',2)),t("p",null,[a[175]||(a[175]=s("We here require that the time step ")),t("mjx-container",$1,[(l(),Q("svg",Y1,a[155]||(a[155]=[e('',1)]))),a[156]||(a[156]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",{mathvariant:"normal"},"Δ"),t("mi",null,"t")])],-1))]),a[176]||(a[176]=s(" is constant. Given the velocity ")),t("mjx-container",_1,[(l(),Q("svg",t2,a[157]||(a[157]=[e('',1)]))),a[158]||(a[158]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"u"),t("mn",null,"0")])])],-1))]),a[177]||(a[177]=s(" and pressure ")),t("mjx-container",a2,[(l(),Q("svg",s2,a[159]||(a[159]=[e('',1)]))),a[160]||(a[160]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 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    julia
    abstract type AbstractRungeKuttaMethod{T} <: IncompressibleNavierStokes.AbstractODEMethod{T}
    Abstract Runge Kutta method.
    Fields
    source
    ',4))]),t("details",j2,[t("summary",null,[a[211]||(a[211]=t("a",{id:"IncompressibleNavierStokes.ExplicitRungeKuttaMethod",href:"#IncompressibleNavierStokes.ExplicitRungeKuttaMethod"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.ExplicitRungeKuttaMethod")],-1)),a[212]||(a[212]=s()),i(T,{type:"info",class:"jlObjectType jlType",text:"Type"})]),a[313]||(a[313]=e('
    julia
    struct ExplicitRungeKuttaMethod{T} <: IncompressibleNavierStokes.AbstractRungeKuttaMethod{T}
    Explicit Runge Kutta method. See Sanderse [17].
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The space-discrete divergence-freeness is thus perfectly preserved, even though the time discretization introduces other errors."))]),a[314]||(a[314]=e('
    Fields
    • A
    • b
    • c
    • r
    • p_add_solve
    source
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    julia
    struct ImplicitRungeKuttaMethod{T} <: IncompressibleNavierStokes.AbstractRungeKuttaMethod{T}
    Implicit Runge Kutta method. See Sanderse [18].
    The implicit linear system is solved at each time step using Newton's method. The newton_type may be one of the following:
    • :no: Replace iteration matrix with I/Δt (no Jacobian)
    • :approximate: Build Jacobian once before iterations only
    • :full: Build Jacobian at each iteration
    Fields
    • A
    • b
    • c
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    • p_add_solve
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    Set up Butcher arrays A, b, and c, as well as and SSP coefficient r. For families of methods, optional input s is the number of stages.
    Original (MATLAB) by David Ketcheson, extended by Benjamin Sanderse.
    Exports
    source
    ',5))]),t("details",P3,[t("summary",null,[a[321]||(a[321]=t("a",{id:"IncompressibleNavierStokes.LMWray3",href:"#IncompressibleNavierStokes.LMWray3"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.LMWray3")],-1)),a[322]||(a[322]=s()),i(T,{type:"info",class:"jlObjectType jlType",text:"Type"})]),a[323]||(a[323]=e('
    julia
    struct LMWray3{T} <: IncompressibleNavierStokes.AbstractRungeKuttaMethod{T}
    Low memory Wray 3rd order scheme. Uses 3 vector fields and one scalar field.
    Fields
    source
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    julia
    FE11(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    FE11 (Forward Euler).
    source
    `,3))]),t("details",z3,[t("summary",null,[a[327]||(a[327]=t("a",{id:"IncompressibleNavierStokes.RKMethods.SSP22",href:"#IncompressibleNavierStokes.RKMethods.SSP22"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.SSP22")],-1)),a[328]||(a[328]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[329]||(a[329]=e(`
    julia
    SSP22(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    FE11 (Forward Euler).
    source
    `,3))]),t("details",z3,[t("summary",null,[a[327]||(a[327]=t("a",{id:"IncompressibleNavierStokes.RKMethods.SSP22",href:"#IncompressibleNavierStokes.RKMethods.SSP22"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.SSP22")],-1)),a[328]||(a[328]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[329]||(a[329]=e(`
    julia
    SSP22(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    SSP22.
    source
    `,3))]),t("details",J3,[t("summary",null,[a[330]||(a[330]=t("a",{id:"IncompressibleNavierStokes.RKMethods.SSP42",href:"#IncompressibleNavierStokes.RKMethods.SSP42"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.SSP42")],-1)),a[331]||(a[331]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[332]||(a[332]=e(`
    julia
    SSP42(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    SSP22.
    source
    `,3))]),t("details",J3,[t("summary",null,[a[330]||(a[330]=t("a",{id:"IncompressibleNavierStokes.RKMethods.SSP42",href:"#IncompressibleNavierStokes.RKMethods.SSP42"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.SSP42")],-1)),a[331]||(a[331]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[332]||(a[332]=e(`
    julia
    SSP42(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    SSP42.
    source
    `,3))]),t("details",U3,[t("summary",null,[a[333]||(a[333]=t("a",{id:"IncompressibleNavierStokes.RKMethods.SSP33",href:"#IncompressibleNavierStokes.RKMethods.SSP33"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.SSP33")],-1)),a[334]||(a[334]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[335]||(a[335]=e(`
    julia
    SSP33(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    SSP42.
    source
    `,3))]),t("details",U3,[t("summary",null,[a[333]||(a[333]=t("a",{id:"IncompressibleNavierStokes.RKMethods.SSP33",href:"#IncompressibleNavierStokes.RKMethods.SSP33"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.SSP33")],-1)),a[334]||(a[334]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[335]||(a[335]=e(`
    julia
    SSP33(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    SSP33.
    source
    `,3))]),t("details",X3,[t("summary",null,[a[336]||(a[336]=t("a",{id:"IncompressibleNavierStokes.RKMethods.SSP43",href:"#IncompressibleNavierStokes.RKMethods.SSP43"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.SSP43")],-1)),a[337]||(a[337]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[338]||(a[338]=e(`
    julia
    SSP43(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    SSP33.
    source
    `,3))]),t("details",X3,[t("summary",null,[a[336]||(a[336]=t("a",{id:"IncompressibleNavierStokes.RKMethods.SSP43",href:"#IncompressibleNavierStokes.RKMethods.SSP43"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.SSP43")],-1)),a[337]||(a[337]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[338]||(a[338]=e(`
    julia
    SSP43(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    SSP43.
    source
    `,3))]),t("details",W3,[t("summary",null,[a[339]||(a[339]=t("a",{id:"IncompressibleNavierStokes.RKMethods.SSP104",href:"#IncompressibleNavierStokes.RKMethods.SSP104"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.SSP104")],-1)),a[340]||(a[340]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[341]||(a[341]=e(`
    julia
    SSP104(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    SSP43.
    source
    `,3))]),t("details",W3,[t("summary",null,[a[339]||(a[339]=t("a",{id:"IncompressibleNavierStokes.RKMethods.SSP104",href:"#IncompressibleNavierStokes.RKMethods.SSP104"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.SSP104")],-1)),a[340]||(a[340]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[341]||(a[341]=e(`
    julia
    SSP104(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    SSP104.
    source
    `,3))]),t("details",q3,[t("summary",null,[a[342]||(a[342]=t("a",{id:"IncompressibleNavierStokes.RKMethods.rSSPs2",href:"#IncompressibleNavierStokes.RKMethods.rSSPs2"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.rSSPs2")],-1)),a[343]||(a[343]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[344]||(a[344]=e(`
    julia
    rSSPs2(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    SSP104.
    source
    `,3))]),t("details",q3,[t("summary",null,[a[342]||(a[342]=t("a",{id:"IncompressibleNavierStokes.RKMethods.rSSPs2",href:"#IncompressibleNavierStokes.RKMethods.rSSPs2"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.rSSPs2")],-1)),a[343]||(a[343]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[344]||(a[344]=e(`
    julia
    rSSPs2(
 ;
     ...
 ) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod, IncompressibleNavierStokes.ImplicitRungeKuttaMethod}
 rSSPs2(
     s;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod, IncompressibleNavierStokes.ImplicitRungeKuttaMethod}
    Rational (optimal, low-storage) s-stage 2nd order SSP.
    source
    `,3))]),t("details",$3,[t("summary",null,[a[345]||(a[345]=t("a",{id:"IncompressibleNavierStokes.RKMethods.rSSPs3",href:"#IncompressibleNavierStokes.RKMethods.rSSPs3"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.rSSPs3")],-1)),a[346]||(a[346]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[347]||(a[347]=e(`
    julia
    rSSPs3(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod, IncompressibleNavierStokes.ImplicitRungeKuttaMethod}
    Rational (optimal, low-storage) s-stage 2nd order SSP.
    source
    `,3))]),t("details",$3,[t("summary",null,[a[345]||(a[345]=t("a",{id:"IncompressibleNavierStokes.RKMethods.rSSPs3",href:"#IncompressibleNavierStokes.RKMethods.rSSPs3"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.rSSPs3")],-1)),a[346]||(a[346]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[347]||(a[347]=e(`
    julia
    rSSPs3(
 ;
     ...
 ) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod, IncompressibleNavierStokes.ImplicitRungeKuttaMethod}
 rSSPs3(
     s;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod, IncompressibleNavierStokes.ImplicitRungeKuttaMethod}
    Rational (optimal, low-storage) s^2-stage 3rd order SSP.
    source
    `,3))]),t("details",Y3,[t("summary",null,[a[348]||(a[348]=t("a",{id:"IncompressibleNavierStokes.RKMethods.Wray3",href:"#IncompressibleNavierStokes.RKMethods.Wray3"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.Wray3")],-1)),a[349]||(a[349]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[350]||(a[350]=e(`
    julia
    Wray3(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod, IncompressibleNavierStokes.ImplicitRungeKuttaMethod}
    Rational (optimal, low-storage) s^2-stage 3rd order SSP.
    source
    `,3))]),t("details",Y3,[t("summary",null,[a[348]||(a[348]=t("a",{id:"IncompressibleNavierStokes.RKMethods.Wray3",href:"#IncompressibleNavierStokes.RKMethods.Wray3"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.Wray3")],-1)),a[349]||(a[349]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[350]||(a[350]=e(`
    julia
    Wray3(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    Wray's RK3.
    source
    `,3))]),t("details",_3,[t("summary",null,[a[351]||(a[351]=t("a",{id:"IncompressibleNavierStokes.RKMethods.RK56",href:"#IncompressibleNavierStokes.RKMethods.RK56"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.RK56")],-1)),a[352]||(a[352]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[353]||(a[353]=e(`
    julia
    RK56(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    Wray's RK3.
    source
    `,3))]),t("details",_3,[t("summary",null,[a[351]||(a[351]=t("a",{id:"IncompressibleNavierStokes.RKMethods.RK56",href:"#IncompressibleNavierStokes.RKMethods.RK56"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.RK56")],-1)),a[352]||(a[352]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[353]||(a[353]=e(`
    julia
    RK56(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    RK56.
    source
    `,3))]),t("details",t4,[t("summary",null,[a[354]||(a[354]=t("a",{id:"IncompressibleNavierStokes.RKMethods.DOPRI6",href:"#IncompressibleNavierStokes.RKMethods.DOPRI6"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.DOPRI6")],-1)),a[355]||(a[355]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[356]||(a[356]=e(`
    julia
    DOPRI6(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    RK56.
    source
    `,3))]),t("details",t4,[t("summary",null,[a[354]||(a[354]=t("a",{id:"IncompressibleNavierStokes.RKMethods.DOPRI6",href:"#IncompressibleNavierStokes.RKMethods.DOPRI6"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.DOPRI6")],-1)),a[355]||(a[355]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[356]||(a[356]=e(`
    julia
    DOPRI6(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    Dormand-Price pair.
    source
    `,3))]),a[483]||(a[483]=t("h3",{id:"Implicit-Methods",tabindex:"-1"},[s("Implicit Methods "),t("a",{class:"header-anchor",href:"#Implicit-Methods","aria-label":'Permalink to "Implicit Methods {#Implicit-Methods}"'},"​")],-1)),t("details",a4,[t("summary",null,[a[357]||(a[357]=t("a",{id:"IncompressibleNavierStokes.RKMethods.BE11",href:"#IncompressibleNavierStokes.RKMethods.BE11"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.BE11")],-1)),a[358]||(a[358]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[359]||(a[359]=e(`
    julia
    BE11(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    Dormand-Price pair.
    source
    `,3))]),a[483]||(a[483]=t("h3",{id:"Implicit-Methods",tabindex:"-1"},[s("Implicit Methods "),t("a",{class:"header-anchor",href:"#Implicit-Methods","aria-label":'Permalink to "Implicit Methods {#Implicit-Methods}"'},"​")],-1)),t("details",a4,[t("summary",null,[a[357]||(a[357]=t("a",{id:"IncompressibleNavierStokes.RKMethods.BE11",href:"#IncompressibleNavierStokes.RKMethods.BE11"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.BE11")],-1)),a[358]||(a[358]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[359]||(a[359]=e(`
    julia
    BE11(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    Backward Euler.
    source
    `,3))]),t("details",s4,[t("summary",null,[a[360]||(a[360]=t("a",{id:"IncompressibleNavierStokes.RKMethods.SDIRK34",href:"#IncompressibleNavierStokes.RKMethods.SDIRK34"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.SDIRK34")],-1)),a[361]||(a[361]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[362]||(a[362]=e(`
    julia
    SDIRK34(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    Backward Euler.
    source
    `,3))]),t("details",s4,[t("summary",null,[a[360]||(a[360]=t("a",{id:"IncompressibleNavierStokes.RKMethods.SDIRK34",href:"#IncompressibleNavierStokes.RKMethods.SDIRK34"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.SDIRK34")],-1)),a[361]||(a[361]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[362]||(a[362]=e(`
    julia
    SDIRK34(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    3-stage, 4th order singly diagonally implicit (SSP).
    source
    `,3))]),t("details",e4,[t("summary",null,[a[363]||(a[363]=t("a",{id:"IncompressibleNavierStokes.RKMethods.ISSPm2",href:"#IncompressibleNavierStokes.RKMethods.ISSPm2"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.ISSPm2")],-1)),a[364]||(a[364]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[365]||(a[365]=e(`
    julia
    ISSPm2(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    3-stage, 4th order singly diagonally implicit (SSP).
    source
    `,3))]),t("details",e4,[t("summary",null,[a[363]||(a[363]=t("a",{id:"IncompressibleNavierStokes.RKMethods.ISSPm2",href:"#IncompressibleNavierStokes.RKMethods.ISSPm2"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.ISSPm2")],-1)),a[364]||(a[364]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[365]||(a[365]=e(`
    julia
    ISSPm2(
 ;
     ...
 ) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
 ISSPm2(
     s;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod, IncompressibleNavierStokes.ImplicitRungeKuttaMethod}
    Optimal DIRK SSP schemes of order 2.
    source
    `,3))]),t("details",Q4,[t("summary",null,[a[366]||(a[366]=t("a",{id:"IncompressibleNavierStokes.RKMethods.ISSPs3",href:"#IncompressibleNavierStokes.RKMethods.ISSPs3"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.ISSPs3")],-1)),a[367]||(a[367]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[368]||(a[368]=e(`
    julia
    ISSPs3(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod, IncompressibleNavierStokes.ImplicitRungeKuttaMethod}
    Optimal DIRK SSP schemes of order 2.
    source
    `,3))]),t("details",Q4,[t("summary",null,[a[366]||(a[366]=t("a",{id:"IncompressibleNavierStokes.RKMethods.ISSPs3",href:"#IncompressibleNavierStokes.RKMethods.ISSPs3"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.ISSPs3")],-1)),a[367]||(a[367]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[368]||(a[368]=e(`
    julia
    ISSPs3(
 ;
     ...
 ) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
 ISSPs3(
     s;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod, IncompressibleNavierStokes.ImplicitRungeKuttaMethod}
    Optimal DIRK SSP schemes of order 3.
    source
    `,3))]),a[484]||(a[484]=t("h3",{id:"Half-explicit-methods",tabindex:"-1"},[s("Half explicit methods "),t("a",{class:"header-anchor",href:"#Half-explicit-methods","aria-label":'Permalink to "Half explicit methods {#Half-explicit-methods}"'},"​")],-1)),t("details",l4,[t("summary",null,[a[369]||(a[369]=t("a",{id:"IncompressibleNavierStokes.RKMethods.HEM3",href:"#IncompressibleNavierStokes.RKMethods.HEM3"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.HEM3")],-1)),a[370]||(a[370]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[371]||(a[371]=e(`
    julia
    HEM3(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod, IncompressibleNavierStokes.ImplicitRungeKuttaMethod}
    Optimal DIRK SSP schemes of order 3.
    source
    `,3))]),a[484]||(a[484]=t("h3",{id:"Half-explicit-methods",tabindex:"-1"},[s("Half explicit methods "),t("a",{class:"header-anchor",href:"#Half-explicit-methods","aria-label":'Permalink to "Half explicit methods {#Half-explicit-methods}"'},"​")],-1)),t("details",l4,[t("summary",null,[a[369]||(a[369]=t("a",{id:"IncompressibleNavierStokes.RKMethods.HEM3",href:"#IncompressibleNavierStokes.RKMethods.HEM3"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.HEM3")],-1)),a[370]||(a[370]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[371]||(a[371]=e(`
    julia
    HEM3(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    Brasey and Hairer.
    source
    `,3))]),t("details",T4,[t("summary",null,[a[372]||(a[372]=t("a",{id:"IncompressibleNavierStokes.RKMethods.HEM3BS",href:"#IncompressibleNavierStokes.RKMethods.HEM3BS"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.HEM3BS")],-1)),a[373]||(a[373]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[374]||(a[374]=e(`
    julia
    HEM3BS(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    Brasey and Hairer.
    source
    `,3))]),t("details",T4,[t("summary",null,[a[372]||(a[372]=t("a",{id:"IncompressibleNavierStokes.RKMethods.HEM3BS",href:"#IncompressibleNavierStokes.RKMethods.HEM3BS"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.HEM3BS")],-1)),a[373]||(a[373]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[374]||(a[374]=e(`
    julia
    HEM3BS(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    HEM3BS.
    source
    `,3))]),t("details",i4,[t("summary",null,[a[375]||(a[375]=t("a",{id:"IncompressibleNavierStokes.RKMethods.HEM5",href:"#IncompressibleNavierStokes.RKMethods.HEM5"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.HEM5")],-1)),a[376]||(a[376]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[377]||(a[377]=e(`
    julia
    HEM5(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    HEM3BS.
    source
    `,3))]),t("details",i4,[t("summary",null,[a[375]||(a[375]=t("a",{id:"IncompressibleNavierStokes.RKMethods.HEM5",href:"#IncompressibleNavierStokes.RKMethods.HEM5"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.HEM5")],-1)),a[376]||(a[376]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[377]||(a[377]=e(`
    julia
    HEM5(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    Brasey and Hairer, 5 stage, 4th order.
    source
    `,3))]),a[485]||(a[485]=t("h3",{id:"Classical-Methods",tabindex:"-1"},[s("Classical Methods "),t("a",{class:"header-anchor",href:"#Classical-Methods","aria-label":'Permalink to "Classical Methods {#Classical-Methods}"'},"​")],-1)),t("details",n4,[t("summary",null,[a[378]||(a[378]=t("a",{id:"IncompressibleNavierStokes.RKMethods.GL1",href:"#IncompressibleNavierStokes.RKMethods.GL1"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.GL1")],-1)),a[379]||(a[379]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[380]||(a[380]=e(`
    julia
    GL1(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    Brasey and Hairer, 5 stage, 4th order.
    source
    `,3))]),a[485]||(a[485]=t("h3",{id:"Classical-Methods",tabindex:"-1"},[s("Classical Methods "),t("a",{class:"header-anchor",href:"#Classical-Methods","aria-label":'Permalink to "Classical Methods {#Classical-Methods}"'},"​")],-1)),t("details",n4,[t("summary",null,[a[378]||(a[378]=t("a",{id:"IncompressibleNavierStokes.RKMethods.GL1",href:"#IncompressibleNavierStokes.RKMethods.GL1"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.GL1")],-1)),a[379]||(a[379]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[380]||(a[380]=e(`
    julia
    GL1(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    GL1.
    source
    `,3))]),t("details",o4,[t("summary",null,[a[381]||(a[381]=t("a",{id:"IncompressibleNavierStokes.RKMethods.GL2",href:"#IncompressibleNavierStokes.RKMethods.GL2"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.GL2")],-1)),a[382]||(a[382]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[383]||(a[383]=e(`
    julia
    GL2(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    GL1.
    source
    `,3))]),t("details",o4,[t("summary",null,[a[381]||(a[381]=t("a",{id:"IncompressibleNavierStokes.RKMethods.GL2",href:"#IncompressibleNavierStokes.RKMethods.GL2"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.GL2")],-1)),a[382]||(a[382]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[383]||(a[383]=e(`
    julia
    GL2(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    GL2.
    source
    `,3))]),t("details",r4,[t("summary",null,[a[384]||(a[384]=t("a",{id:"IncompressibleNavierStokes.RKMethods.GL3",href:"#IncompressibleNavierStokes.RKMethods.GL3"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.GL3")],-1)),a[385]||(a[385]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[386]||(a[386]=e(`
    julia
    GL3(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    GL2.
    source
    `,3))]),t("details",r4,[t("summary",null,[a[384]||(a[384]=t("a",{id:"IncompressibleNavierStokes.RKMethods.GL3",href:"#IncompressibleNavierStokes.RKMethods.GL3"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.GL3")],-1)),a[385]||(a[385]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[386]||(a[386]=e(`
    julia
    GL3(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    GL3.
    source
    `,3))]),t("details",d4,[t("summary",null,[a[387]||(a[387]=t("a",{id:"IncompressibleNavierStokes.RKMethods.RIA1",href:"#IncompressibleNavierStokes.RKMethods.RIA1"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.RIA1")],-1)),a[388]||(a[388]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[389]||(a[389]=e(`
    julia
    RIA1(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    GL3.
    source
    `,3))]),t("details",d4,[t("summary",null,[a[387]||(a[387]=t("a",{id:"IncompressibleNavierStokes.RKMethods.RIA1",href:"#IncompressibleNavierStokes.RKMethods.RIA1"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.RIA1")],-1)),a[388]||(a[388]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[389]||(a[389]=e(`
    julia
    RIA1(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    This is implicit Euler.
    source
    `,3))]),t("details",m4,[t("summary",null,[a[390]||(a[390]=t("a",{id:"IncompressibleNavierStokes.RKMethods.RIA2",href:"#IncompressibleNavierStokes.RKMethods.RIA2"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.RIA2")],-1)),a[391]||(a[391]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[392]||(a[392]=e(`
    julia
    RIA2(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    This is implicit Euler.
    source
    `,3))]),t("details",m4,[t("summary",null,[a[390]||(a[390]=t("a",{id:"IncompressibleNavierStokes.RKMethods.RIA2",href:"#IncompressibleNavierStokes.RKMethods.RIA2"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.RIA2")],-1)),a[391]||(a[391]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[392]||(a[392]=e(`
    julia
    RIA2(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    RIA2.
    source
    `,3))]),t("details",p4,[t("summary",null,[a[393]||(a[393]=t("a",{id:"IncompressibleNavierStokes.RKMethods.RIA3",href:"#IncompressibleNavierStokes.RKMethods.RIA3"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.RIA3")],-1)),a[394]||(a[394]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[395]||(a[395]=e(`
    julia
    RIA3(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    RIA2.
    source
    `,3))]),t("details",p4,[t("summary",null,[a[393]||(a[393]=t("a",{id:"IncompressibleNavierStokes.RKMethods.RIA3",href:"#IncompressibleNavierStokes.RKMethods.RIA3"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.RIA3")],-1)),a[394]||(a[394]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[395]||(a[395]=e(`
    julia
    RIA3(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    RIA3.
    source
    `,3))]),t("details",h4,[t("summary",null,[a[396]||(a[396]=t("a",{id:"IncompressibleNavierStokes.RKMethods.RIIA1",href:"#IncompressibleNavierStokes.RKMethods.RIIA1"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.RIIA1")],-1)),a[397]||(a[397]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[398]||(a[398]=e(`
    julia
    RIIA1(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    RIA3.
    source
    `,3))]),t("details",h4,[t("summary",null,[a[396]||(a[396]=t("a",{id:"IncompressibleNavierStokes.RKMethods.RIIA1",href:"#IncompressibleNavierStokes.RKMethods.RIIA1"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.RIIA1")],-1)),a[397]||(a[397]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[398]||(a[398]=e(`
    julia
    RIIA1(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    RIIA1.
    source
    `,3))]),t("details",g4,[t("summary",null,[a[399]||(a[399]=t("a",{id:"IncompressibleNavierStokes.RKMethods.RIIA2",href:"#IncompressibleNavierStokes.RKMethods.RIIA2"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.RIIA2")],-1)),a[400]||(a[400]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[401]||(a[401]=e(`
    julia
    RIIA2(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    RIIA1.
    source
    `,3))]),t("details",g4,[t("summary",null,[a[399]||(a[399]=t("a",{id:"IncompressibleNavierStokes.RKMethods.RIIA2",href:"#IncompressibleNavierStokes.RKMethods.RIIA2"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.RIIA2")],-1)),a[400]||(a[400]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[401]||(a[401]=e(`
    julia
    RIIA2(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    RIIA2.
    source
    `,3))]),t("details",k4,[t("summary",null,[a[402]||(a[402]=t("a",{id:"IncompressibleNavierStokes.RKMethods.RIIA3",href:"#IncompressibleNavierStokes.RKMethods.RIIA3"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.RIIA3")],-1)),a[403]||(a[403]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[404]||(a[404]=e(`
    julia
    RIIA3(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    RIIA2.
    source
    `,3))]),t("details",k4,[t("summary",null,[a[402]||(a[402]=t("a",{id:"IncompressibleNavierStokes.RKMethods.RIIA3",href:"#IncompressibleNavierStokes.RKMethods.RIIA3"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.RIIA3")],-1)),a[403]||(a[403]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[404]||(a[404]=e(`
    julia
    RIIA3(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    RIIA3.
    source
    `,3))]),t("details",c4,[t("summary",null,[a[405]||(a[405]=t("a",{id:"IncompressibleNavierStokes.RKMethods.LIIIA2",href:"#IncompressibleNavierStokes.RKMethods.LIIIA2"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.LIIIA2")],-1)),a[406]||(a[406]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[407]||(a[407]=e(`
    julia
    LIIIA2(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    RIIA3.
    source
    `,3))]),t("details",c4,[t("summary",null,[a[405]||(a[405]=t("a",{id:"IncompressibleNavierStokes.RKMethods.LIIIA2",href:"#IncompressibleNavierStokes.RKMethods.LIIIA2"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.LIIIA2")],-1)),a[406]||(a[406]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[407]||(a[407]=e(`
    julia
    LIIIA2(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    LIIIA2.
    source
    `,3))]),t("details",u4,[t("summary",null,[a[408]||(a[408]=t("a",{id:"IncompressibleNavierStokes.RKMethods.LIIIA3",href:"#IncompressibleNavierStokes.RKMethods.LIIIA3"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.LIIIA3")],-1)),a[409]||(a[409]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[410]||(a[410]=e(`
    julia
    LIIIA3(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    LIIIA2.
    source
    `,3))]),t("details",u4,[t("summary",null,[a[408]||(a[408]=t("a",{id:"IncompressibleNavierStokes.RKMethods.LIIIA3",href:"#IncompressibleNavierStokes.RKMethods.LIIIA3"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.LIIIA3")],-1)),a[409]||(a[409]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[410]||(a[410]=e(`
    julia
    LIIIA3(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    LIIIA3.
    source
    `,3))]),a[486]||(a[486]=t("h3",{id:"Chebyshev-methods",tabindex:"-1"},[s("Chebyshev methods "),t("a",{class:"header-anchor",href:"#Chebyshev-methods","aria-label":'Permalink to "Chebyshev methods {#Chebyshev-methods}"'},"​")],-1)),t("details",H4,[t("summary",null,[a[411]||(a[411]=t("a",{id:"IncompressibleNavierStokes.RKMethods.CHDIRK3",href:"#IncompressibleNavierStokes.RKMethods.CHDIRK3"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.CHDIRK3")],-1)),a[412]||(a[412]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[413]||(a[413]=e(`
    julia
    CHDIRK3(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    LIIIA3.
    source
    `,3))]),a[486]||(a[486]=t("h3",{id:"Chebyshev-methods",tabindex:"-1"},[s("Chebyshev methods "),t("a",{class:"header-anchor",href:"#Chebyshev-methods","aria-label":'Permalink to "Chebyshev methods {#Chebyshev-methods}"'},"​")],-1)),t("details",H4,[t("summary",null,[a[411]||(a[411]=t("a",{id:"IncompressibleNavierStokes.RKMethods.CHDIRK3",href:"#IncompressibleNavierStokes.RKMethods.CHDIRK3"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.CHDIRK3")],-1)),a[412]||(a[412]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[413]||(a[413]=e(`
    julia
    CHDIRK3(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    Chebyshev based DIRK (not algebraically stable).
    source
    `,3))]),t("details",y4,[t("summary",null,[a[414]||(a[414]=t("a",{id:"IncompressibleNavierStokes.RKMethods.CHCONS3",href:"#IncompressibleNavierStokes.RKMethods.CHCONS3"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.CHCONS3")],-1)),a[415]||(a[415]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[416]||(a[416]=e(`
    julia
    CHCONS3(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    Chebyshev based DIRK (not algebraically stable).
    source
    `,3))]),t("details",y4,[t("summary",null,[a[414]||(a[414]=t("a",{id:"IncompressibleNavierStokes.RKMethods.CHCONS3",href:"#IncompressibleNavierStokes.RKMethods.CHCONS3"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.CHCONS3")],-1)),a[415]||(a[415]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[416]||(a[416]=e(`
    julia
    CHCONS3(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    CHCONS3.
    source
    `,3))]),t("details",w4,[t("summary",null,[a[417]||(a[417]=t("a",{id:"IncompressibleNavierStokes.RKMethods.CHC3",href:"#IncompressibleNavierStokes.RKMethods.CHC3"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.CHC3")],-1)),a[418]||(a[418]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[419]||(a[419]=e(`
    julia
    CHC3(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    CHCONS3.
    source
    `,3))]),t("details",w4,[t("summary",null,[a[417]||(a[417]=t("a",{id:"IncompressibleNavierStokes.RKMethods.CHC3",href:"#IncompressibleNavierStokes.RKMethods.CHC3"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.CHC3")],-1)),a[418]||(a[418]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[419]||(a[419]=e(`
    julia
    CHC3(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    Chebyshev quadrature and C(3) satisfied. Note this equals Lobatto IIIA.
    source
    `,3))]),t("details",x4,[t("summary",null,[a[420]||(a[420]=t("a",{id:"IncompressibleNavierStokes.RKMethods.CHC5",href:"#IncompressibleNavierStokes.RKMethods.CHC5"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.CHC5")],-1)),a[421]||(a[421]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[422]||(a[422]=e(`
    julia
    CHC5(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    Chebyshev quadrature and C(3) satisfied. Note this equals Lobatto IIIA.
    source
    `,3))]),t("details",x4,[t("summary",null,[a[420]||(a[420]=t("a",{id:"IncompressibleNavierStokes.RKMethods.CHC5",href:"#IncompressibleNavierStokes.RKMethods.CHC5"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.CHC5")],-1)),a[421]||(a[421]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[422]||(a[422]=e(`
    julia
    CHC5(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    CHC5.
    source
    `,3))]),a[487]||(a[487]=t("h3",{id:"Miscellaneous-Methods",tabindex:"-1"},[s("Miscellaneous Methods "),t("a",{class:"header-anchor",href:"#Miscellaneous-Methods","aria-label":'Permalink to "Miscellaneous Methods {#Miscellaneous-Methods}"'},"​")],-1)),t("details",f4,[t("summary",null,[a[423]||(a[423]=t("a",{id:"IncompressibleNavierStokes.RKMethods.Mid22",href:"#IncompressibleNavierStokes.RKMethods.Mid22"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.Mid22")],-1)),a[424]||(a[424]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[425]||(a[425]=e(`
    julia
    Mid22(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    CHC5.
    source
    `,3))]),a[487]||(a[487]=t("h3",{id:"Miscellaneous-Methods",tabindex:"-1"},[s("Miscellaneous Methods "),t("a",{class:"header-anchor",href:"#Miscellaneous-Methods","aria-label":'Permalink to "Miscellaneous Methods {#Miscellaneous-Methods}"'},"​")],-1)),t("details",f4,[t("summary",null,[a[423]||(a[423]=t("a",{id:"IncompressibleNavierStokes.RKMethods.Mid22",href:"#IncompressibleNavierStokes.RKMethods.Mid22"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.Mid22")],-1)),a[424]||(a[424]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[425]||(a[425]=e(`
    julia
    Mid22(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    Midpoint 22 method.
    source
    `,3))]),t("details",b4,[t("summary",null,[a[426]||(a[426]=t("a",{id:"IncompressibleNavierStokes.RKMethods.MTE22",href:"#IncompressibleNavierStokes.RKMethods.MTE22"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.MTE22")],-1)),a[427]||(a[427]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[428]||(a[428]=e(`
    julia
    MTE22(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    Midpoint 22 method.
    source
    `,3))]),t("details",b4,[t("summary",null,[a[426]||(a[426]=t("a",{id:"IncompressibleNavierStokes.RKMethods.MTE22",href:"#IncompressibleNavierStokes.RKMethods.MTE22"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.MTE22")],-1)),a[427]||(a[427]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[428]||(a[428]=e(`
    julia
    MTE22(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    Minimal truncation error 22 method (Heun).
    source
    `,3))]),t("details",M4,[t("summary",null,[a[429]||(a[429]=t("a",{id:"IncompressibleNavierStokes.RKMethods.CN22",href:"#IncompressibleNavierStokes.RKMethods.CN22"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.CN22")],-1)),a[430]||(a[430]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[431]||(a[431]=e(`
    julia
    CN22(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    Minimal truncation error 22 method (Heun).
    source
    `,3))]),t("details",M4,[t("summary",null,[a[429]||(a[429]=t("a",{id:"IncompressibleNavierStokes.RKMethods.CN22",href:"#IncompressibleNavierStokes.RKMethods.CN22"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.CN22")],-1)),a[430]||(a[430]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[431]||(a[431]=e(`
    julia
    CN22(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    Crank-Nicholson.
    source
    `,3))]),t("details",L4,[t("summary",null,[a[432]||(a[432]=t("a",{id:"IncompressibleNavierStokes.RKMethods.Heun33",href:"#IncompressibleNavierStokes.RKMethods.Heun33"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.Heun33")],-1)),a[433]||(a[433]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[434]||(a[434]=e(`
    julia
    Heun33(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    Crank-Nicholson.
    source
    `,3))]),t("details",L4,[t("summary",null,[a[432]||(a[432]=t("a",{id:"IncompressibleNavierStokes.RKMethods.Heun33",href:"#IncompressibleNavierStokes.RKMethods.Heun33"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.Heun33")],-1)),a[433]||(a[433]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[434]||(a[434]=e(`
    julia
    Heun33(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    Heun33.
    source
    `,3))]),t("details",v4,[t("summary",null,[a[435]||(a[435]=t("a",{id:"IncompressibleNavierStokes.RKMethods.RK33C2",href:"#IncompressibleNavierStokes.RKMethods.RK33C2"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.RK33C2")],-1)),a[436]||(a[436]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[437]||(a[437]=e(`
    julia
    RK33C2(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    Heun33.
    source
    `,3))]),t("details",v4,[t("summary",null,[a[435]||(a[435]=t("a",{id:"IncompressibleNavierStokes.RKMethods.RK33C2",href:"#IncompressibleNavierStokes.RKMethods.RK33C2"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.RK33C2")],-1)),a[436]||(a[436]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[437]||(a[437]=e(`
    julia
    RK33C2(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    RK3 satisfying C(2) for i=3.
    source
    `,3))]),t("details",E4,[t("summary",null,[a[438]||(a[438]=t("a",{id:"IncompressibleNavierStokes.RKMethods.RK33P2",href:"#IncompressibleNavierStokes.RKMethods.RK33P2"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.RK33P2")],-1)),a[439]||(a[439]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[440]||(a[440]=e(`
    julia
    RK33P2(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    RK3 satisfying C(2) for i=3.
    source
    `,3))]),t("details",E4,[t("summary",null,[a[438]||(a[438]=t("a",{id:"IncompressibleNavierStokes.RKMethods.RK33P2",href:"#IncompressibleNavierStokes.RKMethods.RK33P2"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.RK33P2")],-1)),a[439]||(a[439]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[440]||(a[440]=e(`
    julia
    RK33P2(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    RK3 satisfying the second order condition for the pressure.
    source
    `,3))]),t("details",Z4,[t("summary",null,[a[441]||(a[441]=t("a",{id:"IncompressibleNavierStokes.RKMethods.RK44",href:"#IncompressibleNavierStokes.RKMethods.RK44"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.RK44")],-1)),a[442]||(a[442]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[443]||(a[443]=e(`
    julia
    RK44(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    RK3 satisfying the second order condition for the pressure.
    source
    `,3))]),t("details",Z4,[t("summary",null,[a[441]||(a[441]=t("a",{id:"IncompressibleNavierStokes.RKMethods.RK44",href:"#IncompressibleNavierStokes.RKMethods.RK44"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.RK44")],-1)),a[442]||(a[442]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[443]||(a[443]=e(`
    julia
    RK44(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    Classical fourth order.
    source
    `,3))]),t("details",j4,[t("summary",null,[a[444]||(a[444]=t("a",{id:"IncompressibleNavierStokes.RKMethods.RK44C2",href:"#IncompressibleNavierStokes.RKMethods.RK44C2"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.RK44C2")],-1)),a[445]||(a[445]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[446]||(a[446]=e(`
    julia
    RK44C2(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    Classical fourth order.
    source
    `,3))]),t("details",j4,[t("summary",null,[a[444]||(a[444]=t("a",{id:"IncompressibleNavierStokes.RKMethods.RK44C2",href:"#IncompressibleNavierStokes.RKMethods.RK44C2"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.RK44C2")],-1)),a[445]||(a[445]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[446]||(a[446]=e(`
    julia
    RK44C2(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    RK4 satisfying C(2) for i=3.
    source
    `,3))]),t("details",D4,[t("summary",null,[a[447]||(a[447]=t("a",{id:"IncompressibleNavierStokes.RKMethods.RK44C23",href:"#IncompressibleNavierStokes.RKMethods.RK44C23"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.RK44C23")],-1)),a[448]||(a[448]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[449]||(a[449]=e(`
    julia
    RK44C23(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    RK4 satisfying C(2) for i=3.
    source
    `,3))]),t("details",D4,[t("summary",null,[a[447]||(a[447]=t("a",{id:"IncompressibleNavierStokes.RKMethods.RK44C23",href:"#IncompressibleNavierStokes.RKMethods.RK44C23"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.RK44C23")],-1)),a[448]||(a[448]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[449]||(a[449]=e(`
    julia
    RK44C23(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    RK4 satisfying C(2) for i=3 and c2=c3.
    source
    `,3))]),t("details",S4,[t("summary",null,[a[450]||(a[450]=t("a",{id:"IncompressibleNavierStokes.RKMethods.RK44P2",href:"#IncompressibleNavierStokes.RKMethods.RK44P2"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.RK44P2")],-1)),a[451]||(a[451]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[452]||(a[452]=e(`
    julia
    RK44P2(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    RK4 satisfying C(2) for i=3 and c2=c3.
    source
    `,3))]),t("details",S4,[t("summary",null,[a[450]||(a[450]=t("a",{id:"IncompressibleNavierStokes.RKMethods.RK44P2",href:"#IncompressibleNavierStokes.RKMethods.RK44P2"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.RK44P2")],-1)),a[451]||(a[451]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[452]||(a[452]=e(`
    julia
    RK44P2(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    RK4 satisfying the second order condition for the pressure (but not third order).
    source
    `,3))]),a[488]||(a[488]=t("h3",{id:"DSRK-Methods",tabindex:"-1"},[s("DSRK Methods "),t("a",{class:"header-anchor",href:"#DSRK-Methods","aria-label":'Permalink to "DSRK Methods {#DSRK-Methods}"'},"​")],-1)),t("details",V4,[t("summary",null,[a[453]||(a[453]=t("a",{id:"IncompressibleNavierStokes.RKMethods.DSso2",href:"#IncompressibleNavierStokes.RKMethods.DSso2"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.DSso2")],-1)),a[454]||(a[454]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[455]||(a[455]=e(`
    julia
    DSso2(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    RK4 satisfying the second order condition for the pressure (but not third order).
    source
    `,3))]),a[488]||(a[488]=t("h3",{id:"DSRK-Methods",tabindex:"-1"},[s("DSRK Methods "),t("a",{class:"header-anchor",href:"#DSRK-Methods","aria-label":'Permalink to "DSRK Methods {#DSRK-Methods}"'},"​")],-1)),t("details",V4,[t("summary",null,[a[453]||(a[453]=t("a",{id:"IncompressibleNavierStokes.RKMethods.DSso2",href:"#IncompressibleNavierStokes.RKMethods.DSso2"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.DSso2")],-1)),a[454]||(a[454]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[455]||(a[455]=e(`
    julia
    DSso2(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    CBM's DSRKso2.
    source
    `,3))]),t("details",I4,[t("summary",null,[a[456]||(a[456]=t("a",{id:"IncompressibleNavierStokes.RKMethods.DSRK2",href:"#IncompressibleNavierStokes.RKMethods.DSRK2"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.DSRK2")],-1)),a[457]||(a[457]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[458]||(a[458]=e(`
    julia
    DSRK2(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    CBM's DSRKso2.
    source
    `,3))]),t("details",I4,[t("summary",null,[a[456]||(a[456]=t("a",{id:"IncompressibleNavierStokes.RKMethods.DSRK2",href:"#IncompressibleNavierStokes.RKMethods.DSRK2"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.DSRK2")],-1)),a[457]||(a[457]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[458]||(a[458]=e(`
    julia
    DSRK2(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    CBM's DSRK2.
    source
    `,3))]),t("details",F4,[t("summary",null,[a[459]||(a[459]=t("a",{id:"IncompressibleNavierStokes.RKMethods.DSRK3",href:"#IncompressibleNavierStokes.RKMethods.DSRK3"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.DSRK3")],-1)),a[460]||(a[460]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[461]||(a[461]=e(`
    julia
    DSRK3(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    CBM's DSRK2.
    source
    `,3))]),t("details",F4,[t("summary",null,[a[459]||(a[459]=t("a",{id:"IncompressibleNavierStokes.RKMethods.DSRK3",href:"#IncompressibleNavierStokes.RKMethods.DSRK3"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.DSRK3")],-1)),a[460]||(a[460]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[461]||(a[461]=e(`
    julia
    DSRK3(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    Zennaro's DSRK3.
    source
    `,3))]),a[489]||(a[489]=t("h3",{id:"Non-SSP-Methods-of-Wong-and-Spiteri",tabindex:"-1"},[s('"Non-SSP" Methods of Wong & Spiteri '),t("a",{class:"header-anchor",href:"#Non-SSP-Methods-of-Wong-and-Spiteri","aria-label":'Permalink to ""Non-SSP" Methods of Wong & Spiteri {#"Non-SSP"-Methods-of-Wong-and-Spiteri}"'},"​")],-1)),t("details",R4,[t("summary",null,[a[462]||(a[462]=t("a",{id:"IncompressibleNavierStokes.RKMethods.NSSP21",href:"#IncompressibleNavierStokes.RKMethods.NSSP21"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.NSSP21")],-1)),a[463]||(a[463]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[464]||(a[464]=e(`
    julia
    NSSP21(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    Zennaro's DSRK3.
    source
    `,3))]),a[489]||(a[489]=t("h3",{id:"Non-SSP-Methods-of-Wong-and-Spiteri",tabindex:"-1"},[s('"Non-SSP" Methods of Wong & Spiteri '),t("a",{class:"header-anchor",href:"#Non-SSP-Methods-of-Wong-and-Spiteri","aria-label":'Permalink to ""Non-SSP" Methods of Wong & Spiteri {#"Non-SSP"-Methods-of-Wong-and-Spiteri}"'},"​")],-1)),t("details",R4,[t("summary",null,[a[462]||(a[462]=t("a",{id:"IncompressibleNavierStokes.RKMethods.NSSP21",href:"#IncompressibleNavierStokes.RKMethods.NSSP21"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.NSSP21")],-1)),a[463]||(a[463]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[464]||(a[464]=e(`
    julia
    NSSP21(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    NSSP21.
    source
    `,3))]),t("details",C4,[t("summary",null,[a[465]||(a[465]=t("a",{id:"IncompressibleNavierStokes.RKMethods.NSSP32",href:"#IncompressibleNavierStokes.RKMethods.NSSP32"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.NSSP32")],-1)),a[466]||(a[466]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[467]||(a[467]=e(`
    julia
    NSSP32(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    NSSP21.
    source
    `,3))]),t("details",C4,[t("summary",null,[a[465]||(a[465]=t("a",{id:"IncompressibleNavierStokes.RKMethods.NSSP32",href:"#IncompressibleNavierStokes.RKMethods.NSSP32"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.NSSP32")],-1)),a[466]||(a[466]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[467]||(a[467]=e(`
    julia
    NSSP32(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    NSSP32.
    source
    `,3))]),t("details",N4,[t("summary",null,[a[468]||(a[468]=t("a",{id:"IncompressibleNavierStokes.RKMethods.NSSP33",href:"#IncompressibleNavierStokes.RKMethods.NSSP33"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.NSSP33")],-1)),a[469]||(a[469]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[470]||(a[470]=e(`
    julia
    NSSP33(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    NSSP32.
    source
    `,3))]),t("details",N4,[t("summary",null,[a[468]||(a[468]=t("a",{id:"IncompressibleNavierStokes.RKMethods.NSSP33",href:"#IncompressibleNavierStokes.RKMethods.NSSP33"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.NSSP33")],-1)),a[469]||(a[469]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[470]||(a[470]=e(`
    julia
    NSSP33(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    NSSP33.
    source
    `,3))]),t("details",K4,[t("summary",null,[a[471]||(a[471]=t("a",{id:"IncompressibleNavierStokes.RKMethods.NSSP53",href:"#IncompressibleNavierStokes.RKMethods.NSSP53"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.NSSP53")],-1)),a[472]||(a[472]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[473]||(a[473]=e(`
    julia
    NSSP53(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    NSSP33.
    source
    `,3))]),t("details",K4,[t("summary",null,[a[471]||(a[471]=t("a",{id:"IncompressibleNavierStokes.RKMethods.NSSP53",href:"#IncompressibleNavierStokes.RKMethods.NSSP53"},[t("span",{class:"jlbinding"},"IncompressibleNavierStokes.RKMethods.NSSP53")],-1)),a[472]||(a[472]=s()),i(T,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[473]||(a[473]=e(`
    julia
    NSSP53(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    NSSP53.
    source
    `,3))])])}const X4=n(r,[["render",A4]]);export{U4 as __pageData,X4 as default};
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    NSSP53.
    source
    `,3))])])}const X4=n(r,[["render",A4]]);export{U4 as __pageData,X4 as default};
diff --git a/dev/assets/manual_utils.md.CBLFMcXv.js b/dev/assets/manual_utils.md.BH_DwoJc.js
similarity index 94%
rename from dev/assets/manual_utils.md.CBLFMcXv.js
rename to dev/assets/manual_utils.md.BH_DwoJc.js
index efe57f223..cc950eac7 100644
--- a/dev/assets/manual_utils.md.CBLFMcXv.js
+++ b/dev/assets/manual_utils.md.BH_DwoJc.js
@@ -1,11 +1,11 @@
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diff --git a/dev/assets/manual_utils.md.CBLFMcXv.lean.js b/dev/assets/manual_utils.md.BH_DwoJc.lean.js
similarity index 94%
rename from dev/assets/manual_utils.md.CBLFMcXv.lean.js
rename to dev/assets/manual_utils.md.BH_DwoJc.lean.js
index efe57f223..cc950eac7 100644
--- a/dev/assets/manual_utils.md.CBLFMcXv.lean.js
+++ b/dev/assets/manual_utils.md.BH_DwoJc.lean.js
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similarity index 100%
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diff --git a/dev/assets/pqbunkl.lOgoLSNC.png b/dev/assets/pqbunkl.lOgoLSNC.png
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    This page was generated using Literate.jl.
    - + \ No newline at end of file diff --git a/dev/examples/generated/BackwardFacingStep2D.html b/dev/examples/generated/BackwardFacingStep2D.html index db24054e1..869b067fe 100644 --- a/dev/examples/generated/BackwardFacingStep2D.html +++ b/dev/examples/generated/BackwardFacingStep2D.html @@ -9,9 +9,9 @@ - + - + @@ -119,7 +119,7 @@ fieldplot(state; setup, size = (600, 150), fieldname = :velocitynorm) fieldplot(state; setup, size = (600, 150), fieldname = :vorticity)

    This page was generated using Literate.jl.

    - + \ No newline at end of file diff --git a/dev/examples/generated/BackwardFacingStep3D.html b/dev/examples/generated/BackwardFacingStep3D.html index 95fdbd923..a171a8db3 100644 --- a/dev/examples/generated/BackwardFacingStep3D.html +++ b/dev/examples/generated/BackwardFacingStep3D.html @@ -9,9 +9,9 @@ - + - + @@ -122,7 +122,7 @@ fieldplot(state; setup, fieldname = :velocitynorm) fieldplot(state; setup, fieldname = :vorticity)

    This page was generated using Literate.jl.

    - + \ No newline at end of file diff --git a/dev/examples/generated/DecayingTurbulence2D.html b/dev/examples/generated/DecayingTurbulence2D.html index 54c3f5356..9d169a6d3 100644 --- a/dev/examples/generated/DecayingTurbulence2D.html +++ b/dev/examples/generated/DecayingTurbulence2D.html @@ -9,11 +9,11 @@ - + - + - + @@ -47,13 +47,14 @@ ), log = timelogger(; nupdate = 100), ), -);
    [ Info: t = 0.0997245	Δt = 0.0011	umax = 3.3	itertime = 0.028
-[ Info: t = 0.205518	Δt = 0.0012	umax = 3	itertime = 0.016
-[ Info: t = 0.311364	Δt = 0.001	umax = 3.5	itertime = 0.016
-[ Info: t = 0.431183	Δt = 0.0013	umax = 2.6	itertime = 0.016
-[ Info: t = 0.579297	Δt = 0.0017	umax = 2.1	itertime = 0.016
-[ Info: t = 0.734226	Δt = 0.0014	umax = 2.4	itertime = 0.016
-[ Info: t = 0.873193	Δt = 0.0014	umax = 2.5	itertime = 0.016

    Post-process ​

    We may visualize or export the computed fields

    Energy history

    julia
    outputs.ehist

    Energy spectrum

    julia
    outputs.espec

    Plot field

    julia
    fieldplot(state; setup)

    Copy-pasteable code ​

    Below is the full code for this example stripped of comments and output.

    julia
    using GLMakie
+);
    [ Info: t = 0.105436	Δt = 0.0011	umax = 3.2	itertime = 0.026
+[ Info: t = 0.216847	Δt = 0.00098	umax = 3.6	itertime = 0.015
+[ Info: t = 0.339039	Δt = 0.0013	umax = 2.6	itertime = 0.015
+[ Info: t = 0.462407	Δt = 0.0012	umax = 3	itertime = 0.015
+[ Info: t = 0.583124	Δt = 0.0012	umax = 2.9	itertime = 0.015
+[ Info: t = 0.720378	Δt = 0.0014	umax = 2.6	itertime = 0.015
+[ Info: t = 0.846368	Δt = 0.0012	umax = 2.9	itertime = 0.015
+[ Info: t = 0.976317	Δt = 0.0014	umax = 2.5	itertime = 0.015

    Post-process ​

    We may visualize or export the computed fields

    Energy history

    julia
    outputs.ehist

    Energy spectrum

    julia
    outputs.espec

    Plot field

    julia
    fieldplot(state; setup)

    Copy-pasteable code ​

    Below is the full code for this example stripped of comments and output.

    julia
    using GLMakie
 using IncompressibleNavierStokes
 
 n = 256
@@ -88,7 +89,7 @@
 outputs.espec
 
 fieldplot(state; setup)

    This page was generated using Literate.jl.

    - + \ No newline at end of file diff --git a/dev/examples/generated/DecayingTurbulence3D.html b/dev/examples/generated/DecayingTurbulence3D.html index c42445a26..974437709 100644 --- a/dev/examples/generated/DecayingTurbulence3D.html +++ b/dev/examples/generated/DecayingTurbulence3D.html @@ -9,9 +9,9 @@ - + - + @@ -80,7 +80,7 @@ # outputs.ehist outputs.espec

    This page was generated using Literate.jl.

    - + \ No newline at end of file diff --git a/dev/examples/generated/Kolmogorov2D.html b/dev/examples/generated/Kolmogorov2D.html index e5ffbf410..888f6b963 100644 --- a/dev/examples/generated/Kolmogorov2D.html +++ b/dev/examples/generated/Kolmogorov2D.html @@ -9,9 +9,9 @@ - + - + @@ -98,7 +98,7 @@ outputs.ehist outputs.espec

    This page was generated using Literate.jl.

    - + \ No newline at end of file diff --git a/dev/examples/generated/LidDrivenCavity2D.html b/dev/examples/generated/LidDrivenCavity2D.html index 437a2dd66..da1c4e661 100644 --- a/dev/examples/generated/LidDrivenCavity2D.html +++ b/dev/examples/generated/LidDrivenCavity2D.html @@ -9,9 +9,9 @@ - + - + @@ -114,7 +114,7 @@ # outputs.vtk # outputs.field outputs.log

    This page was generated using Literate.jl.

    - + \ No newline at end of file diff --git a/dev/examples/generated/LidDrivenCavity3D.html b/dev/examples/generated/LidDrivenCavity3D.html index e6144b112..1beedde01 100644 --- a/dev/examples/generated/LidDrivenCavity3D.html +++ b/dev/examples/generated/LidDrivenCavity3D.html @@ -9,9 +9,9 @@ - + - + @@ -102,7 +102,7 @@ save_vtk(state; setup, filename = joinpath(outdir, "solution")) outputs.ehist

    This page was generated using Literate.jl.

    - + \ No newline at end of file diff --git a/dev/examples/generated/MultiActuator.html b/dev/examples/generated/MultiActuator.html index 4797d983e..a9c10d0a4 100644 --- a/dev/examples/generated/MultiActuator.html +++ b/dev/examples/generated/MultiActuator.html @@ -9,9 +9,9 @@ - + - + @@ -216,7 +216,7 @@ fig = fieldplot(state; setup, size = (600, 300), fieldname = :vorticity) lines!.(boxes; color = :red); fig

    This page was generated using Literate.jl.

    - + \ No newline at end of file diff --git a/dev/examples/generated/PlanarMixing2D.html b/dev/examples/generated/PlanarMixing2D.html index 11f2068cb..104a21439 100644 --- a/dev/examples/generated/PlanarMixing2D.html +++ b/dev/examples/generated/PlanarMixing2D.html @@ -9,9 +9,9 @@ - + - + @@ -129,7 +129,7 @@ outputs.rtp save_vtk(state; setup, filename = joinpath(outdir, "solution"), psolver)

    This page was generated using Literate.jl.

    - + \ No newline at end of file diff --git a/dev/examples/generated/PlaneJets2D.html b/dev/examples/generated/PlaneJets2D.html index fccff6437..e23d71a65 100644 --- a/dev/examples/generated/PlaneJets2D.html +++ b/dev/examples/generated/PlaneJets2D.html @@ -9,9 +9,9 @@ - + - + @@ -296,7 +296,7 @@ fieldplot(state; setup, fieldname = :velocitynorm) fieldplot(state; setup, fieldname = :vorticity)

    This page was generated using Literate.jl.

    - + \ No newline at end of file diff --git a/dev/examples/generated/RayleighBenard2D.html b/dev/examples/generated/RayleighBenard2D.html index 0bb3e9f50..a619eddb8 100644 --- a/dev/examples/generated/RayleighBenard2D.html +++ b/dev/examples/generated/RayleighBenard2D.html @@ -9,11 +9,11 @@ - + - + - + @@ -84,7 +84,7 @@ nondim_type = 1, )
    (α1 = 0.00026645826f0, α2 = 1.0f0, α3 = 0.00026645826f0, α4 = 0.0003752933f0, γ = 1.0f0, dodissipation = true, boundary_conditions = ((SymmetricBC(), SymmetricBC()), (DirichletBC{Float32}(1.0f0), DirichletBC{Float32}(0.0f0))), gdir = 2)

    Grid

    julia
    n = 100
 x = tanh_grid(T(0), T(2), 2n, T(1.2)), tanh_grid(T(0), T(1), n, T(1.2))
-plotgrid(x...)

    Setup

    julia
    setup = Setup(;
+plotgrid(x...)

    Setup

    julia
    setup = Setup(;
     x,
     boundary_conditions = ((DirichletBC(), DirichletBC()), (DirichletBC(), DirichletBC())),
     Re = 1 / temperature.α1,
@@ -118,8 +118,8 @@
     tlims = (T(0), T(20)),
     Δt = T(1e-2),
     processors,
-);
    [ Info: t = 10.0001	Δt = 0.01	umax = 0.24	itertime = 0.016
-[ Info: t = 20.0004	Δt = 0.01	umax = 0.55	itertime = 0.013

    Nusselt numbers

    julia
    outputs.nusselt

    Average temperature

    julia
    outputs.avg

    Copy-pasteable code ​

    Below is the full code for this example stripped of comments and output.

    julia
    using GLMakie
+);
    [ Info: t = 10.0001	Δt = 0.01	umax = 0.24	itertime = 0.015
+[ Info: t = 20.0004	Δt = 0.01	umax = 0.55	itertime = 0.012

    Nusselt numbers

    julia
    outputs.nusselt

    Average temperature

    julia
    outputs.avg

    Copy-pasteable code ​

    Below is the full code for this example stripped of comments and output.

    julia
    using GLMakie
 using IncompressibleNavierStokes
 
 outdir = joinpath(@__DIR__, "output", "RayleighBenard2D")
@@ -245,7 +245,7 @@
 outputs.nusselt
 
 outputs.avg

    This page was generated using Literate.jl.

    - + \ No newline at end of file diff --git a/dev/examples/generated/RayleighBenard3D.html b/dev/examples/generated/RayleighBenard3D.html index 3f1ea4b41..a2d77722f 100644 --- a/dev/examples/generated/RayleighBenard3D.html +++ b/dev/examples/generated/RayleighBenard3D.html @@ -9,9 +9,9 @@ - + - + @@ -197,7 +197,7 @@ # fieldname = :eig2field, fieldname = :temperature, )

    This page was generated using Literate.jl.

    - + \ No newline at end of file diff --git a/dev/examples/generated/RayleighTaylor2D.html b/dev/examples/generated/RayleighTaylor2D.html index 04c8e0c4e..659185a2b 100644 --- a/dev/examples/generated/RayleighTaylor2D.html +++ b/dev/examples/generated/RayleighTaylor2D.html @@ -9,11 +9,11 @@ - + - + - + @@ -29,7 +29,7 @@ # using CUDA, CUDSS # backend = CUDABackend()
    CPU(false)

    Precision

    julia
    T = Float64
    Float64

    Grid

    julia
    n = 50
 x = tanh_grid(T(0), T(1), n, T(1.5)), tanh_grid(T(0), T(2), 2n, T(1.5))
-plotgrid(x...; figure = (; size = (300, 600)))

    Setup

    julia
    temperature = temperature_equation(;
+plotgrid(x...; figure = (; size = (300, 600)))

    Setup

    julia
    temperature = temperature_equation(;
     Pr = T(0.71),
     Ra = T(1e6),
     Ge = T(1.0),
@@ -60,14 +60,14 @@
         log = timelogger(; nupdate = 200),
     ),
 );
    [ Info: t = 1	Δt = 0.005	umax = 0.1	itertime = 0.011
-[ Info: t = 2	Δt = 0.005	umax = 0.32	itertime = 0.0033
-[ Info: t = 3	Δt = 0.005	umax = 0.78	itertime = 0.0034
-[ Info: t = 4	Δt = 0.005	umax = 0.75	itertime = 0.0033
-[ Info: t = 5	Δt = 0.005	umax = 0.78	itertime = 0.0033
+[ Info: t = 2	Δt = 0.005	umax = 0.32	itertime = 0.0032
+[ Info: t = 3	Δt = 0.005	umax = 0.78	itertime = 0.0032
+[ Info: t = 4	Δt = 0.005	umax = 0.75	itertime = 0.0032
+[ Info: t = 5	Δt = 0.005	umax = 0.78	itertime = 0.0032
 [ Info: t = 6	Δt = 0.005	umax = 0.84	itertime = 0.0033
-[ Info: t = 7	Δt = 0.005	umax = 0.84	itertime = 0.0033
-[ Info: t = 8	Δt = 0.005	umax = 0.89	itertime = 0.0033
-[ Info: t = 9	Δt = 0.005	umax = 1.2	itertime = 0.0033
+[ Info: t = 7	Δt = 0.005	umax = 0.84	itertime = 0.0032
+[ Info: t = 8	Δt = 0.005	umax = 0.89	itertime = 0.0032
+[ Info: t = 9	Δt = 0.005	umax = 1.2	itertime = 0.0032
 [ Info: t = 10	Δt = 0.005	umax = 0.88	itertime = 0.0033

    Copy-pasteable code ​

    Below is the full code for this example stripped of comments and output.

    julia
    using GLMakie
 using IncompressibleNavierStokes
 
@@ -119,7 +119,7 @@
         log = timelogger(; nupdate = 200),
     ),
 );

    This page was generated using Literate.jl.

    - + \ No newline at end of file diff --git a/dev/examples/generated/RayleighTaylor3D.html b/dev/examples/generated/RayleighTaylor3D.html index 025a526cc..16a5d06c6 100644 --- a/dev/examples/generated/RayleighTaylor3D.html +++ b/dev/examples/generated/RayleighTaylor3D.html @@ -9,9 +9,9 @@ - + - + @@ -184,7 +184,7 @@ hist(vec(Array(log.(max.(eps(T), .-field))))) fieldplot(state; setup, fieldname = :temperature)

    This page was generated using Literate.jl.

    - + \ No newline at end of file diff --git a/dev/examples/generated/ShearLayer2D.html b/dev/examples/generated/ShearLayer2D.html index 1a0723ca4..e519f6077 100644 --- a/dev/examples/generated/ShearLayer2D.html +++ b/dev/examples/generated/ShearLayer2D.html @@ -9,9 +9,9 @@ - + - + @@ -107,7 +107,7 @@ fieldplot(state; setup, fieldname = :velocitynorm) fieldplot(state; setup, fieldname = :vorticity)

    This page was generated using Literate.jl.

    - + \ No newline at end of file diff --git a/dev/examples/generated/TaylorGreenVortex2D.html b/dev/examples/generated/TaylorGreenVortex2D.html index 9ee7e1fdd..bceff0c9c 100644 --- a/dev/examples/generated/TaylorGreenVortex2D.html +++ b/dev/examples/generated/TaylorGreenVortex2D.html @@ -9,11 +9,11 @@ - + - + - + @@ -88,7 +88,7 @@ scatterlines!(ax, nlist, e; label = "Data") lines!(ax, collect(extrema(nlist)), n -> n^-2.0; linestyle = :dash, label = "n^-2") axislegend(ax) -fig

    Save figure

    julia
    save(joinpath(outdir, "convergence.png"), fig)
    CairoMakie.Screen{IMAGE}

    Copy-pasteable code ​

    Below is the full code for this example stripped of comments and output.

    julia
    using GLMakie
+fig

    Save figure

    julia
    save(joinpath(outdir, "convergence.png"), fig)
    CairoMakie.Screen{IMAGE}

    Copy-pasteable code ​

    Below is the full code for this example stripped of comments and output.

    julia
    using GLMakie
 using IncompressibleNavierStokes
 using LinearAlgebra
 
@@ -158,7 +158,7 @@
 fig
 
 save(joinpath(outdir, "convergence.png"), fig)

    This page was generated using Literate.jl.

    - + \ No newline at end of file diff --git a/dev/examples/generated/TaylorGreenVortex3D.html b/dev/examples/generated/TaylorGreenVortex3D.html index 33e6d2dbf..48567f89e 100644 --- a/dev/examples/generated/TaylorGreenVortex3D.html +++ b/dev/examples/generated/TaylorGreenVortex3D.html @@ -9,9 +9,9 @@ - + - + @@ -105,7 +105,7 @@ outputs.ehist outputs.espec

    This page was generated using Literate.jl.

    - + \ No newline at end of file diff --git a/dev/examples/index.html b/dev/examples/index.html index 95add314c..037494d34 100644 --- a/dev/examples/index.html +++ b/dev/examples/index.html @@ -9,9 +9,9 @@ - + - + @@ -24,7 +24,7 @@
    IncompressibleNavierStokes.jl

    Appearance

    Examples gallery ​

    Here is a gallery of selected commented example simulations. The outputs are generated with Literate.jl and displayed inline. Copy-pasteable code is available at the bottom of each example. The raw Julia source scripts can be found in the examples folder.

    Simple flows ​

    Decaying turbulence (2D)

    Decaying turbulence in a periodic 2D-box

    Decaying turbulence (3D)

    Decaying turbulence in a periodic 3D-box

    Taylor-Green vortex (2D)

    Decaying vortex structures in a periodic 2D-box

    Taylor-Green vortex (3D)

    Decaying vortex structures in a periodic 3D-box

    Kolmogorov flow (2D)

    Initiate a flow through a sinusoidal force

    Shear-layer (2D)

    Two layers with different velocities

    Planar mixing (2D)

    Planar mixing layers

    Flows with mixed boundary conditions ​

    Actuator (2D)

    Unsteady inflow around an actuator disk

    Actuator (3D)

    Unsteady inflow around an actuator disk

    Backward Facing Step (2D)

    Flow past a backward facing step

    Backward Facing Step (3D)

    Flow past a backward facing step

    Lid-driven cavity (2D)

    Generate a flow caused by a moving lid

    Lid-driven cavity (3D)

    Generate a flow caused by a moving lid

    Unsteady inflow around multiple actuator disks

    Multi-actuator (2D)

    With a temperature equation ​

    Rayleigh-Bénard convection (2D)

    Convection generated by a temperature gradient between a hot and a cold plate

    Rayleigh-Bénard convection (3D)

    Convection generated by a temperature gradient between a hot and a cold plate

    Rayleigh-Taylor instability (2D)

    Convection generated by a temperature gradient between hot and cold fluids

    Rayleigh-Taylor instability (3D)

    Convection generated by a temperature gradient between hot and cold fluids

    Neural network closure models ​

    Filter analysis

    Compare discrete filters and their properties with DNS data

    CNN closure models

    Compare CNN closure models for different filters, grid sizes, and projection orders

    Symmetry-preserving closure models

    Train group equivariant CNNs and compare to normal CNNs

    - + \ No newline at end of file diff --git a/dev/getting_started.html b/dev/getting_started.html index f8f6c1838..98a2ff6c8 100644 --- a/dev/getting_started.html +++ b/dev/getting_started.html @@ -9,9 +9,9 @@ - + - + @@ -24,7 +24,7 @@
    IncompressibleNavierStokes.jl

    Appearance

    Getting Started ​

    To install IncompressibleNavierStokes, open up a Julia-REPL, type ] to get into Pkg-mode, and type:

    sh
    add IncompressibleNavierStokes

    which will install the package and all dependencies to your local environment. Note that IncompressibleNavierStokes requires Julia version 1.9 or above.

    Check out the gallery for commented example simulations.

    - + \ No newline at end of file diff --git a/dev/hashmap.json b/dev/hashmap.json index bb3bd2cd4..fed04badb 100644 --- a/dev/hashmap.json +++ b/dev/hashmap.json @@ -1 +1 @@ -{"about_citing.md":"_SqI95YY","about_contributing.md":"DhMuxJGd","about_development.md":"y-9cwpPK","about_index.md":"BYzFEErW","about_license.md":"BMTPbAIg","about_versions.md":"ByqO0kAn","examples_generated_actuator2d.md":"CicJvifW","examples_generated_actuator3d.md":"m7eZ843V","examples_generated_backwardfacingstep2d.md":"FwRjGPk6","examples_generated_backwardfacingstep3d.md":"Bt6sInAp","examples_generated_decayingturbulence2d.md":"BUaxapZ_","examples_generated_decayingturbulence3d.md":"6T1PfoFX","examples_generated_kolmogorov2d.md":"B1tvst8N","examples_generated_liddrivencavity2d.md":"DUDWAH6I","examples_generated_liddrivencavity3d.md":"3_GDZ_fM","examples_generated_multiactuator.md":"DMsLnVhM","examples_generated_planarmixing2d.md":"DZ7I9yl7","examples_generated_planejets2d.md":"Db3tovgb","examples_generated_rayleighbenard2d.md":"Cmmj9k1P","examples_generated_rayleighbenard3d.md":"DctDp5W_","examples_generated_rayleightaylor2d.md":"C4WmLQbK","examples_generated_rayleightaylor3d.md":"Egg4joXV","examples_generated_shearlayer2d.md":"05unvAuP","examples_generated_taylorgreenvortex2d.md":"XLjKfMqJ","examples_generated_taylorgreenvortex3d.md":"m01ixPJ8","examples_index.md":"CqFB2DAD","getting_started.md":"BC8qOMfm","index.md":"D9tgK7Nd","manual_closure.md":"Bl_bxqy7","manual_differentiability.md":"DHl4CtMi","manual_gpu.md":"sbkSDAqo","manual_les.md":"O3bTXciX","manual_matrices.md":"CgitaAGh","manual_ns.md":"iA_pjZRQ","manual_operators.md":"d5WWXEo-","manual_precision.md":"b0OoKca_","manual_pressure.md":"CRdAlBfk","manual_sciml.md":"CER13ld6","manual_setup.md":"w6x6ynsm","manual_solver.md":"CSATxmgf","manual_spatial.md":"BY8cpZ4v","manual_temperature.md":"CvylYS1B","manual_time.md":"3eTgRN26","manual_utils.md":"CBLFMcXv","references.md":"DpSG3aTl"} +{"about_citing.md":"_SqI95YY","about_contributing.md":"DhMuxJGd","about_development.md":"y-9cwpPK","about_index.md":"DDeqRwIY","about_license.md":"B43Z_-HQ","about_versions.md":"Bk-sITSg","examples_generated_actuator2d.md":"DxDFVKMk","examples_generated_actuator3d.md":"m7eZ843V","examples_generated_backwardfacingstep2d.md":"FwRjGPk6","examples_generated_backwardfacingstep3d.md":"Bt6sInAp","examples_generated_decayingturbulence2d.md":"Di45Ivji","examples_generated_decayingturbulence3d.md":"6T1PfoFX","examples_generated_kolmogorov2d.md":"B1tvst8N","examples_generated_liddrivencavity2d.md":"DUDWAH6I","examples_generated_liddrivencavity3d.md":"3_GDZ_fM","examples_generated_multiactuator.md":"DMsLnVhM","examples_generated_planarmixing2d.md":"DZ7I9yl7","examples_generated_planejets2d.md":"Db3tovgb","examples_generated_rayleighbenard2d.md":"DK8wXqn9","examples_generated_rayleighbenard3d.md":"DctDp5W_","examples_generated_rayleightaylor2d.md":"QTiLYjii","examples_generated_rayleightaylor3d.md":"Egg4joXV","examples_generated_shearlayer2d.md":"05unvAuP","examples_generated_taylorgreenvortex2d.md":"BBlb1Y2_","examples_generated_taylorgreenvortex3d.md":"m01ixPJ8","examples_index.md":"CqFB2DAD","getting_started.md":"BC8qOMfm","index.md":"D9tgK7Nd","manual_closure.md":"Bp4YxggZ","manual_differentiability.md":"D_UkzJTY","manual_gpu.md":"sbkSDAqo","manual_les.md":"O3bTXciX","manual_matrices.md":"WgRK4vkk","manual_ns.md":"iA_pjZRQ","manual_operators.md":"DDNX2oaK","manual_precision.md":"b0OoKca_","manual_pressure.md":"CGOE2Wqh","manual_sciml.md":"CcNBgquF","manual_setup.md":"B7b6UkIH","manual_solver.md":"CveGplQk","manual_spatial.md":"BY8cpZ4v","manual_temperature.md":"DGNhfuFk","manual_time.md":"BJBCC_BN","manual_utils.md":"BH_DwoJc","references.md":"DpSG3aTl"} diff --git a/dev/index.html b/dev/index.html index 6d8efa466..a5a522a99 100644 --- a/dev/index.html +++ b/dev/index.html @@ -9,9 +9,9 @@ - + - + @@ -24,7 +24,7 @@ - + \ No newline at end of file diff --git a/dev/manual/closure.html b/dev/manual/closure.html index e8b37e025..a80e6cf58 100644 --- a/dev/manual/closure.html +++ b/dev/manual/closure.html @@ -9,11 +9,11 @@ - + - + - + @@ -24,10 +24,10 @@
    IncompressibleNavierStokes.jl

    Appearance

    Neural closure models ​

    NeuralClosure

    These features are experimental, and require cloning IncompressibleNavierStokes from GitHub:

    sh
    git clone https://github.com/agdestein/IncompressibleNavierStokes.jl
-cd IncompressibleNavierStokes/lib/NeuralClosure

    Large eddy simulation, a closure model is required. With IncompressibleNavierStokes, a neural closure model can be trained on filtered DNS data. The discrete DNS equations are given by

    Mu=0,dudt=F(u)−Gp.

    Applying a spatial filter Φ, the extracted large scale components u¯=Φu are governed by the equation

    Mu¯=0,du¯dt=F(u¯)+c−Gp¯,

    where the discretizations M, F, and G are adapted to the size of their inputs and c=F(u)―−F(u¯) is a commutator error. We here assumed that M and Φ commute, which is the case for face averaging filters. Replacing c with a parameterized closure model m(u¯,θ)≈c gives the LES equations for the approximate large scale velocity v¯≈u¯

    Mv¯=0,dv¯dt=F(v¯)+m(v¯,θ)−Gq¯.

    NeuralClosure module ​

    IncompressibleNavierStokes provides a NeuralClosure module.

    NeuralClosure.NeuralClosure Module

    Neural closure modelling tools.

    source

    NeuralClosure.collocate Method
    julia
    collocate(u) -> Any

    Interpolate velocity components to volume centers.

    source

    NeuralClosure.create_closure Method
    julia
    create_closure(layers...; rng)

    Create neural closure model from layers.

    source

    NeuralClosure.create_tensorclosure Method
    julia
    create_tensorclosure(layers...; setup, rng)

    Create tensor basis closure.

    source

    NeuralClosure.decollocate Method
    julia
    decollocate(u) -> Any

    Interpolate closure force from volume centers to volume faces.

    source

    NeuralClosure.wrappedclosure Method
    julia
    wrappedclosure(
+cd IncompressibleNavierStokes/lib/NeuralClosure

    Large eddy simulation, a closure model is required. With IncompressibleNavierStokes, a neural closure model can be trained on filtered DNS data. The discrete DNS equations are given by

    Mu=0,dudt=F(u)−Gp.

    Applying a spatial filter Φ, the extracted large scale components u¯=Φu are governed by the equation

    Mu¯=0,du¯dt=F(u¯)+c−Gp¯,

    where the discretizations M, F, and G are adapted to the size of their inputs and c=F(u)―−F(u¯) is a commutator error. We here assumed that M and Φ commute, which is the case for face averaging filters. Replacing c with a parameterized closure model m(u¯,θ)≈c gives the LES equations for the approximate large scale velocity v¯≈u¯

    Mv¯=0,dv¯dt=F(v¯)+m(v¯,θ)−Gq¯.

    NeuralClosure module ​

    IncompressibleNavierStokes provides a NeuralClosure module.

    NeuralClosure.NeuralClosure Module

    Neural closure modelling tools.

    source

    NeuralClosure.collocate Method
    julia
    collocate(u) -> Any

    Interpolate velocity components to volume centers.

    source

    NeuralClosure.create_closure Method
    julia
    create_closure(layers...; rng)

    Create neural closure model from layers.

    source

    NeuralClosure.create_tensorclosure Method
    julia
    create_tensorclosure(layers...; setup, rng)

    Create tensor basis closure.

    source

    NeuralClosure.decollocate Method
    julia
    decollocate(u) -> Any

    Interpolate closure force from volume centers to volume faces.

    source

    NeuralClosure.wrappedclosure Method
    julia
    wrappedclosure(
     m,
     setup
-) -> NeuralClosure.var"#neuralclosure#1"

    Wrap closure model and parameters so that it can be used in the solver.

    source

    Filters ​

    The following filters are available:

    NeuralClosure.AbstractFilter Type
    julia
    abstract type AbstractFilter

    Discrete DNS filter.

    Subtypes ConcreteFilter should implement the in-place method:

    (::ConcreteFilter)(v, u, setup_les, compression)

    which filters the DNS field u and put result in LES field v. Then the out-of place method:

    (::ConcreteFilter)(u, setup_les, compression)

    automatically becomes available.

    Fields

    source

    NeuralClosure.FaceAverage Type
    julia
    struct FaceAverage <: NeuralClosure.AbstractFilter

    Average fine grid velocity field over coarse volume face.

    Fields

    source

    NeuralClosure.VolumeAverage Type
    julia
    struct VolumeAverage <: NeuralClosure.AbstractFilter

    Average fine grid velocity field over coarse volume.

    Fields

    source

    NeuralClosure.reconstruct! Method
    julia
    reconstruct!(u, v, setup_dns, setup_les, comp) -> Any

    Reconstruct DNS velocity u from LES velocity v.

    source

    NeuralClosure.reconstruct Method
    julia
    reconstruct(v, setup_dns, setup_les, comp) -> Any

    Reconstruct DNS velocity field. See also reconstruct!.

    source

    Training ​

    To improve the model parameters, we exploit exact filtered DNS data u¯ and exact commutator errors c obtained through DNS. The model is trained by minimizing the a priori loss function

    Lprior(θ)=∥m(u¯,θ)−c∥2,

    or the a posteriori loss function

    Lpost(θ)=∥v¯θ−u¯∥2,

    where v¯θ is the solution to the LES equation for the given parameters θ. The prior loss is easy to evaluate and easy to differentiate, as it does not involve solving the ODE. However, minimizing Lprior does not take into account the effect of the prediction error on the LES solution error. The posterior loss does, but has a longer computational chain involving solving the LES ODE.

    NeuralClosure.create_callback Method
    julia
    create_callback(
+) -> NeuralClosure.var"#neuralclosure#1"

    Wrap closure model and parameters so that it can be used in the solver.

    source

    Filters ​

    The following filters are available:

    NeuralClosure.AbstractFilter Type
    julia
    abstract type AbstractFilter

    Discrete DNS filter.

    Subtypes ConcreteFilter should implement the in-place method:

    (::ConcreteFilter)(v, u, setup_les, compression)

    which filters the DNS field u and put result in LES field v. Then the out-of place method:

    (::ConcreteFilter)(u, setup_les, compression)

    automatically becomes available.

    Fields

    source

    NeuralClosure.FaceAverage Type
    julia
    struct FaceAverage <: NeuralClosure.AbstractFilter

    Average fine grid velocity field over coarse volume face.

    Fields

    source

    NeuralClosure.VolumeAverage Type
    julia
    struct VolumeAverage <: NeuralClosure.AbstractFilter

    Average fine grid velocity field over coarse volume.

    Fields

    source

    NeuralClosure.reconstruct! Method
    julia
    reconstruct!(u, v, setup_dns, setup_les, comp) -> Any

    Reconstruct DNS velocity u from LES velocity v.

    source

    NeuralClosure.reconstruct Method
    julia
    reconstruct(v, setup_dns, setup_les, comp) -> Any

    Reconstruct DNS velocity field. See also reconstruct!.

    source

    Training ​

    To improve the model parameters, we exploit exact filtered DNS data u¯ and exact commutator errors c obtained through DNS. The model is trained by minimizing the a priori loss function

    Lprior(θ)=∥m(u¯,θ)−c∥2,

    or the a posteriori loss function

    Lpost(θ)=∥v¯θ−u¯∥2,

    where v¯θ is the solution to the LES equation for the given parameters θ. The prior loss is easy to evaluate and easy to differentiate, as it does not involve solving the ODE. However, minimizing Lprior does not take into account the effect of the prediction error on the LES solution error. The posterior loss does, but has a longer computational chain involving solving the LES ODE.

    NeuralClosure.create_callback Method
    julia
    create_callback(
     err;
     θ,
     callbackstate,
@@ -36,29 +36,29 @@
     displayupdates,
     figfile,
     nupdate
-)

    Create convergence plot for relative error between f(x, θ) and y. At each callback, plot is updated and current error is printed.

    If state is nonempty, it also plots previous convergence.

    If not using interactive GLMakie window, set displayupdates to true.

    source

    NeuralClosure.create_dataloader_post Method
    julia
    create_dataloader_post(
+)

    Create convergence plot for relative error between f(x, θ) and y. At each callback, plot is updated and current error is printed.

    If state is nonempty, it also plots previous convergence.

    If not using interactive GLMakie window, set displayupdates to true.

    source

    NeuralClosure.create_dataloader_post Method
    julia
    create_dataloader_post(
     trajectories;
     ntrajectory,
     nunroll,
     device
-)

    Create trajectory dataloader.

    source

    NeuralClosure.create_dataloader_prior Method
    julia
    create_dataloader_prior(
+)

    Create trajectory dataloader.

    source

    NeuralClosure.create_dataloader_prior Method
    julia
    create_dataloader_prior(
     data;
     batchsize,
     device
-) -> NeuralClosure.var"#dataloader#44"{Int64}

    Create dataloader that uses a batch of batchsize random samples from data at each evaluation. The batch is moved to device.

    source

    NeuralClosure.create_loss_post Method
    julia
    create_loss_post(
+) -> NeuralClosure.var"#dataloader#44"{Int64}

    Create dataloader that uses a batch of batchsize random samples from data at each evaluation. The batch is moved to device.

    source

    NeuralClosure.create_loss_post Method
    julia
    create_loss_post(
 ;
     setup,
     method,
     psolver,
     closure,
     nsubstep
-)

    Create a-posteriori loss function.

    source

    NeuralClosure.create_loss_prior Function
    julia
    create_loss_prior(
+)

    Create a-posteriori loss function.

    source

    NeuralClosure.create_loss_prior Function
    julia
    create_loss_prior(
     f
 ) -> NeuralClosure.var"#loss_prior#57"{_A, NeuralClosure.var"#55#56"} where _A
 create_loss_prior(
     f,
     normalize
-) -> NeuralClosure.var"#loss_prior#57"

    Return mean squared error loss for the predictor f.

    source

    NeuralClosure.create_relerr_post Method
    julia
    create_relerr_post(
+) -> NeuralClosure.var"#loss_prior#57"

    Return mean squared error loss for the predictor f.

    source

    NeuralClosure.create_relerr_post Method
    julia
    create_relerr_post(
 ;
     data,
     setup,
@@ -66,7 +66,7 @@
     psolver,
     closure_model,
     nsubstep
-)

    Create a-posteriori relative error.

    source

    NeuralClosure.create_relerr_prior Method
    julia
    create_relerr_prior(f, x, y) -> NeuralClosure.var"#58#59"

    Create a-priori error.

    source

    NeuralClosure.create_relerr_symmetry_post Method
    julia
    create_relerr_symmetry_post(
+)

    Create a-posteriori relative error.

    source

    NeuralClosure.create_relerr_prior Method
    julia
    create_relerr_prior(f, x, y) -> NeuralClosure.var"#58#59"

    Create a-priori error.

    source

    NeuralClosure.create_relerr_symmetry_post Method
    julia
    create_relerr_symmetry_post(
 ;
     u,
     setup,
@@ -75,7 +75,7 @@
     Δt,
     nstep,
     g
-)

    Create a-posteriori symmetry error.

    source

    NeuralClosure.create_relerr_symmetry_prior Method
    julia
    create_relerr_symmetry_prior(; u, setup, g)

    Create a-priori equivariance error.

    source

    NeuralClosure.train Method
    julia
    train(
+)

    Create a-posteriori symmetry error.

    source

    NeuralClosure.create_relerr_symmetry_prior Method
    julia
    create_relerr_symmetry_prior(; u, setup, g)

    Create a-priori equivariance error.

    source

    NeuralClosure.train Method
    julia
    train(
 ;
     dataloader,
     loss,
@@ -84,7 +84,7 @@
     callback,
     callbackstate,
     λ
-)

    Update parameters θ to minimize loss(dataloader(), θ) using the optimiser opt for niter iterations.

    Return the a new named tuple (; opt, θ, callbackstate) with updated state and parameters.

    source

    NeuralClosure.trainepoch Method
    julia
    trainepoch(
+)

    Update parameters θ to minimize loss(dataloader(), θ) using the optimiser opt for niter iterations.

    Return the a new named tuple (; opt, θ, callbackstate) with updated state and parameters.

    source

    NeuralClosure.trainepoch Method
    julia
    trainepoch(
 ;
     dataloader,
     loss,
@@ -94,7 +94,7 @@
     device,
     noiselevel,
     λ
-)

    Update parameters θ to minimize loss(dataloader(), θ) using the optimiser opt for niter iterations.

    Return the a new named tuple (; opt, θ, callbackstate) with updated state and parameters.

    source

    Neural architectures ​

    We provide neural architectures: A convolutional neural network (CNN), group convolutional neural networks (G-CNN) and a Fourier neural operator (FNO).

    NeuralClosure.cnn Method
    julia
    cnn(
+)

    Update parameters θ to minimize loss(dataloader(), θ) using the optimiser opt for niter iterations.

    Return the a new named tuple (; opt, θ, callbackstate) with updated state and parameters.

    source

    Neural architectures ​

    We provide neural architectures: A convolutional neural network (CNN), group convolutional neural networks (G-CNN) and a Fourier neural operator (FNO).

    NeuralClosure.cnn Method
    julia
    cnn(
 ;
     setup,
     radii,
@@ -103,7 +103,7 @@
     use_bias,
     channel_augmenter,
     rng
-)

    Create CNN closure model. Return a tuple (closure, θ) where θ are the initial parameters and closure(u, θ) predicts the commutator error.

    source

    NeuralClosure.GroupConv2D Type
    julia
    struct GroupConv2D{C} <: LuxCore.AbstractLuxLayer

    Group-equivariant convolutional layer – with respect to the p4 group. The layer is equivariant to rotations and translations of the input vector field.

    The kwargs are passed to the Conv layer.

    The layer has three variants:

    • If islifting then it lifts a vector input (u1, u2) into a rotation-state vector (v1, v2, v3, v4).

    • If isprojecting, it projects a rotation-state vector (u1, u2, u3, v4) into a vector (v1, v2).

    • Otherwise, it cyclically transforms the rotation-state vector (u1, u2, u3, u4) into a new rotation-state vector (v1, v2, v3, v4).

    Fields

    • islifting

    • isprojecting

    • cin

    • cout

    • conv

    source

    NeuralClosure.gcnn Method
    julia
    gcnn(; setup, radii, channels, activations, use_bias, rng)

    Create CNN closure model. Return a tuple (closure, θ) where θ are the initial parameters and closure(u, θ) predicts the commutator error.

    source

    NeuralClosure.rot2 Function
    julia
    rot2(u, r)

    Rotate the field u by 90 degrees counter-clockwise r - 1 times.

    source

    NeuralClosure.rot2 Method
    julia
    rot2(u::Tuple{T, T}, r) -> Union{Nothing, Tuple{Any, Any}}

    Rotate vector fields [ux;;; uy]

    source

    NeuralClosure.rot2stag Method
    julia
    rot2stag(u, g) -> Any

    Rotate staggered grid velocity field. See also rot2.

    source

    NeuralClosure.vecrot2 Method
    julia
    vecrot2(u, r) -> Any

    Rotate vector fields [ux;;; uy]

    source

    NeuralClosure.FourierLayer Type
    julia
    struct FourierLayer{D, A, F} <: LuxCore.AbstractLuxLayer

    Fourier layer operating on uniformly discretized functions.

    Some important sizes:

    • dimension: Spatial dimension, e.g. Dimension(2) or Dimension(3).

    • (nx..., cin, nsample): Input size

    • (nx..., cout, nsample): Output size

    • nx = fill(n, dimension()): Number of points in each spatial dimension

    • n ≥ kmax: Same number of points in each spatial dimension, must be larger than cut-off wavenumber

    • kmax: Cut-off wavenumber

    • nsample: Number of input samples (treated independently)

    Fields

    • dimension

    • kmax

    • cin

    • cout

    • σ

    • init_weight

    source

    NeuralClosure.fno Method
    julia
    fno(; setup, kmax, c, σ, ψ, rng, kwargs...)

    Create FNO closure model. Return a tuple (closure, θ) where θ are the initial parameters and closure(V, θ) predicts the commutator error.

    source

    Data generation ​

    NeuralClosure.create_io_arrays Method
    julia
    create_io_arrays(data, setup) -> NamedTuple

    Create (u¯,c) pairs for a-priori training.

    source

    NeuralClosure.create_les_data Method
    julia
    create_les_data(
+)

    Create CNN closure model. Return a tuple (closure, θ) where θ are the initial parameters and closure(u, θ) predicts the commutator error.

    source

    NeuralClosure.GroupConv2D Type
    julia
    struct GroupConv2D{C} <: LuxCore.AbstractLuxLayer

    Group-equivariant convolutional layer – with respect to the p4 group. The layer is equivariant to rotations and translations of the input vector field.

    The kwargs are passed to the Conv layer.

    The layer has three variants:

    • If islifting then it lifts a vector input (u1, u2) into a rotation-state vector (v1, v2, v3, v4).

    • If isprojecting, it projects a rotation-state vector (u1, u2, u3, v4) into a vector (v1, v2).

    • Otherwise, it cyclically transforms the rotation-state vector (u1, u2, u3, u4) into a new rotation-state vector (v1, v2, v3, v4).

    Fields

    • islifting

    • isprojecting

    • cin

    • cout

    • conv

    source

    NeuralClosure.gcnn Method
    julia
    gcnn(; setup, radii, channels, activations, use_bias, rng)

    Create CNN closure model. Return a tuple (closure, θ) where θ are the initial parameters and closure(u, θ) predicts the commutator error.

    source

    NeuralClosure.rot2 Function
    julia
    rot2(u, r)

    Rotate the field u by 90 degrees counter-clockwise r - 1 times.

    source

    NeuralClosure.rot2 Method
    julia
    rot2(u::Tuple{T, T}, r) -> Union{Nothing, Tuple{Any, Any}}

    Rotate vector fields [ux;;; uy]

    source

    NeuralClosure.rot2stag Method
    julia
    rot2stag(u, g) -> Any

    Rotate staggered grid velocity field. See also rot2.

    source

    NeuralClosure.vecrot2 Method
    julia
    vecrot2(u, r) -> Any

    Rotate vector fields [ux;;; uy]

    source

    NeuralClosure.FourierLayer Type
    julia
    struct FourierLayer{D, A, F} <: LuxCore.AbstractLuxLayer

    Fourier layer operating on uniformly discretized functions.

    Some important sizes:

    • dimension: Spatial dimension, e.g. Dimension(2) or Dimension(3).

    • (nx..., cin, nsample): Input size

    • (nx..., cout, nsample): Output size

    • nx = fill(n, dimension()): Number of points in each spatial dimension

    • n ≥ kmax: Same number of points in each spatial dimension, must be larger than cut-off wavenumber

    • kmax: Cut-off wavenumber

    • nsample: Number of input samples (treated independently)

    Fields

    • dimension

    • kmax

    • cin

    • cout

    • σ

    • init_weight

    source

    NeuralClosure.fno Method
    julia
    fno(; setup, kmax, c, σ, ψ, rng, kwargs...)

    Create FNO closure model. Return a tuple (closure, θ) where θ are the initial parameters and closure(V, θ) predicts the commutator error.

    source

    Data generation ​

    NeuralClosure.create_io_arrays Method
    julia
    create_io_arrays(data, setup) -> NamedTuple

    Create (u¯,c) pairs for a-priori training.

    source

    NeuralClosure.create_les_data Method
    julia
    create_les_data(
 ;
     D,
     Re,
@@ -123,7 +123,7 @@
     rng,
     filenames,
     kwargs...
-)

    Create filtered DNS data.

    source

    NeuralClosure.filtersaver Method
    julia
    filtersaver(
+)

    Create filtered DNS data.

    source

    NeuralClosure.filtersaver Method
    julia
    filtersaver(
     dns,
     les,
     filters,
@@ -134,8 +134,8 @@
     filenames,
     F,
     p
-)

    Save filtered DNS data.

    source

    - +)

    Save filtered DNS data.

    source

    + \ No newline at end of file diff --git a/dev/manual/differentiability.html b/dev/manual/differentiability.html index 3ddad1bc4..b691ec5ee 100644 --- a/dev/manual/differentiability.html +++ b/dev/manual/differentiability.html @@ -9,11 +9,11 @@ - + - + - + @@ -113,10 +113,10 @@ (dft-vrank>=1-x51/1 (dft-ct-dit/10 (dftw-direct-10/6 "t3fv_10_avx2_128") - (dft-directbuf/14-10-x10 "n1fv_10_avx2")))), [6.945445343448e-310 6.9454449186883e-310 … 5.0e-324 5.0e-324; 6.9454449184535e-310 6.94544491869066e-310 … 6.94544489383323e-310 6.9454448939123e-310; … ; 6.94544491868354e-310 6.9454449003201e-310 … 5.0e-324 5.0e-324; 6.9454449186859e-310 6.9454449003225e-310 … 6.9454448939107e-310 6.94544489398975e-310], ComplexF64[6.94557386934597e-310 + 6.94557386934597e-310im 0.0 + 0.0im … 0.0 + 0.0im 0.0 + 0.0im; 6.94557386934597e-310 + 6.94557386934597e-310im 0.0 + 0.0im … 0.0 + 0.0im 0.0 + 0.0im; … ; 0.0 + 0.0im 0.0 + 0.0im … 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im … 0.0 + 0.0im 0.0 + 0.0im], Core.Box(([0.0, 0.003946543143456876, 0.01577059737104434, 0.035425498542622634, 0.06283367774273778, 0.09788696740969285, 0.1404470282234972, 0.19034589506796099, 0.24738663991227286, 0.3113441489959698 … 3.68865585100403, 3.752613360087727, 3.809654104932039, 3.859552971776503, 3.9021130325903073, 3.9371663222572626, 3.9645745014573777, 3.984229402628956, 3.996053456856544, 4.0], [0.0, 0.003946543143456876, 0.01577059737104434, 0.035425498542622634, 0.06283367774273778, 0.09788696740969285, 0.1404470282234972, 0.19034589506796099, 0.24738663991227286, 0.3113441489959698 … 0.3819660112501049, 0.3113441489959696, 0.24738663991227255, 0.19034589506796068, 0.14044702822349744, 0.09788696740969302, 0.06283367774273789, 0.0354254985426227, 0.015770597371044366, 0.003946543143456883])), 2, CartesianIndices((2:101, 2:101)))])

    Now, we call the autodiff function from Enzyme:

    julia
    Enzyme.autodiff(Enzyme.Reverse, f!, Duplicated(dudt,ddudt), Duplicated(u,du), Const(params_ref), Const(t))
    ((nothing, nothing, nothing, nothing),)

    Since we have passed a Duplicated object, the gradient of u is stored in du.

    Finally, we can also compare its value with the one obtained by Zygote differentiating the out-of-place (non-mutating) version of the right-hand side:

    julia
    f = create_right_hand_side(setup, psolver)
+      (dft-directbuf/14-10-x10 "n1fv_10_avx2")))), [3.86215878e-315 0.0 … 6.548379943e-315 3.89362187e-316; 6.547993426e-315 0.0 … 6.548380615e-315 3.89362187e-316; … ; 4.245092511e-314 3.22322206e-315 … 6.34907335e-315 0.0; 0.0 6.91711403163036e-310 … 2.9e-322 6.34907303e-315], ComplexF64[1.89668687e-315 + 2.20435554e-315im 5.0e-324 + 6.91697151413794e-310im … 1.191723655e-315 + 4.9e-322im 3.0355e-320 + 6.37e-322im; 1.89668687e-315 + 2.20435554e-315im 5.0e-324 + 6.91697151413952e-310im … 3.9e-322 + 5.0e-324im 5.3e-322 + 3.05534496e-315im; … ; 5.0e-324 + 6.9169715141348e-310im 5.0e-324 + 6.9169715142154e-310im … 5.24e-322 + 2.3716543e-316im 6.4e-322 + 3.955624e-316im; 5.0e-324 + 6.91697151413636e-310im 5.0e-324 + 6.916971514217e-310im … 6.4e-322 + 5.3e-322im 7.9e-322 + 6.47e-322im], Core.Box(([0.0, 0.003946543143456876, 0.01577059737104434, 0.035425498542622634, 0.06283367774273778, 0.09788696740969285, 0.1404470282234972, 0.19034589506796099, 0.24738663991227286, 0.3113441489959698  …  3.68865585100403, 3.752613360087727, 3.809654104932039, 3.859552971776503, 3.9021130325903073, 3.9371663222572626, 3.9645745014573777, 3.984229402628956, 3.996053456856544, 4.0], [0.0, 0.003946543143456876, 0.01577059737104434, 0.035425498542622634, 0.06283367774273778, 0.09788696740969285, 0.1404470282234972, 0.19034589506796099, 0.24738663991227286, 0.3113441489959698  …  0.3819660112501049, 0.3113441489959696, 0.24738663991227255, 0.19034589506796068, 0.14044702822349744, 0.09788696740969302, 0.06283367774273789, 0.0354254985426227, 0.015770597371044366, 0.003946543143456883])), 2, CartesianIndices((2:101, 2:101)))])

    Now, we call the autodiff function from Enzyme:

    julia
    Enzyme.autodiff(Enzyme.Reverse, f!, Duplicated(dudt,ddudt), Duplicated(u,du), Const(params_ref), Const(t))
    ((nothing, nothing, nothing, nothing),)

    Since we have passed a Duplicated object, the gradient of u is stored in du.

    Finally, we can also compare its value with the one obtained by Zygote differentiating the out-of-place (non-mutating) version of the right-hand side:

    julia
    f = create_right_hand_side(setup, psolver)
 _, zpull = Zygote.pullback(f, u, nothing, 0.0);
 @assert zpull(dudt)[1] == du
    - + \ No newline at end of file diff --git a/dev/manual/gpu.html b/dev/manual/gpu.html index 247ae326c..04782f281 100644 --- a/dev/manual/gpu.html +++ b/dev/manual/gpu.html @@ -9,9 +9,9 @@ - + - + @@ -25,7 +25,7 @@
    IncompressibleNavierStokes.jl

    Appearance

    GPU Support ​

    IncompressibleNavierStokes supports various array types. The desired backend only has to be passed to the Setup function:

    julia
    using CUDA
 setup = Setup(; kwargs..., backend = CUDABackend())

    All operators have been made are backend agnostic by using KernelAbstractions.jl. Even if a GPU is not available, the operators are multithreaded if Julia is started with multiple threads (e.g. julia -t 4)

    • This has been tested with CUDA compatible GPUs.

    • This has not been tested with other GPU interfaces, such as

      If they start supporting sparse matrices and fast Fourier transforms they could also be used.

    psolver_direct on CUDA

    To use a specialized linear solver for CUDA, make sure to install and using CUDA.jl and CUDSS.jl. Then psolver_direct will automatically use the CUDSS solver.

    - + \ No newline at end of file diff --git a/dev/manual/les.html b/dev/manual/les.html index 7dd621c6c..9ebeee47f 100644 --- a/dev/manual/les.html +++ b/dev/manual/les.html @@ -9,9 +9,9 @@ - + - + @@ -24,7 +24,7 @@
    IncompressibleNavierStokes.jl

    Appearance

    Large eddy simulation ​

    Depending on the problem specification, a given grid resolution may not be sufficient to resolve all spatial features of the flow. Consider the following example:

    On the left, the grid spacing is too large to capt the smallest eddies in the flow. These eddies create sub-grid stresses that also affect the large scale features. The grid must be refined if we want to compute these stresses exactly.

    On the right, the smallest spatial feature of the flow is fully resolved, and there are no sub-grid stresses. The equations can be solved without worrying about errors from unresolved features. This is known as Direct Numerical Simulation (DNS).

    If refining the grid is too costly, a closure model can be used to predict the sub-grid stresses. The models only give an estimate for these stresses, and may need to be calibrated to the given problem. When used correctly, they can predict the evolution of the large fluid motions without computing the sub-grid motions themselves. This is known as Large Eddy Simulation (LES).

    Eddy viscosity models add a local contribution to the global baseline viscosity. The baseline viscosity models transfer of energy from resolved to atomic scales. The new turbulent viscosity on the other hand, models energy transfer from resolved to unresolved scales. This non-constant field is computed from the local velocity field.

    - + \ No newline at end of file diff --git a/dev/manual/matrices.html b/dev/manual/matrices.html index 99b7a9715..08d1f0766 100644 --- a/dev/manual/matrices.html +++ b/dev/manual/matrices.html @@ -9,11 +9,11 @@ - + - + - + @@ -101,7 +101,7 @@ D = diffusion_mat(setup) d_kernel = diffusion(apply_bc_u(u, 0.0, setup), setup; use_viscosity = false) d_matrix = reshape(D * B * u[:], size(u)) -maximum(abs, d_matrix - d_kernel)
    9.094947017729282e-13

    Matrices only work on flattened fields u[:], while the kernels work on (D+1)-array-shaped fields for a dimension D∈{2,3}.

    Boundary conditions and matrices ​

    Matrices can only be used to represent boundary conditions that depend linearly on the input, such as periodic or Neumann boundary conditions. Non-zero Dirichlet boundary conditions need to be accounted for separately. Consider the following inflow-setup:

    julia
    setup = Setup(;
+maximum(abs, d_matrix - d_kernel)
    4.547473508864641e-13

    Matrices only work on flattened fields u[:], while the kernels work on (D+1)-array-shaped fields for a dimension D∈{2,3}.

    Boundary conditions and matrices ​

    Matrices can only be used to represent boundary conditions that depend linearly on the input, such as periodic or Neumann boundary conditions. Non-zero Dirichlet boundary conditions need to be accounted for separately. Consider the following inflow-setup:

    julia
    setup = Setup(;
     x = (ax, ax),
     boundary_conditions = (
         (DirichletBC((10.0, 0.0)), PressureBC()),
@@ -133,16 +133,16 @@
  0.0     0.0     0.0     0.0     0.0        0.0     0.0     0.0     0.0  0.0
  0.0     0.0     0.0     0.0     0.0  …     0.0     0.0     0.0     0.0  0.0
  0.0     0.0     0.0     0.0     0.0        0.0     0.0     0.0     0.0  0.0
- 0.0     0.0     0.0     0.0     0.0        0.0     0.0     0.0     0.0  0.0

    By adding yM, we get the equality:

    julia
    maximum(abs, (div_matrix + yM) - div_kernel)
    2.842170943040401e-14

    API ​

    IncompressibleNavierStokes.bc_p_mat Function

    Matrix for applying boundary conditions to pressure fields p.

    source

    IncompressibleNavierStokes.bc_temp_mat Function

    Matrix for applying boundary conditions to temperature fields temp.

    source

    IncompressibleNavierStokes.bc_u_mat Function

    Create matrix for applying boundary conditions to velocity fields u. This matrix only applies the boundary conditions depending on u itself (e.g. PeriodicBC). It does not apply constant boundary conditions (e.g. non-zero DirichletBC).

    source

    IncompressibleNavierStokes.diffusion_mat Method
    julia
    diffusion_mat(
+ 0.0     0.0     0.0     0.0     0.0        0.0     0.0     0.0     0.0  0.0

    By adding yM, we get the equality:

    julia
    maximum(abs, (div_matrix + yM) - div_kernel)
    2.842170943040401e-14

    API ​

    IncompressibleNavierStokes.bc_p_mat Function

    Matrix for applying boundary conditions to pressure fields p.

    source

    IncompressibleNavierStokes.bc_temp_mat Function

    Matrix for applying boundary conditions to temperature fields temp.

    source

    IncompressibleNavierStokes.bc_u_mat Function

    Create matrix for applying boundary conditions to velocity fields u. This matrix only applies the boundary conditions depending on u itself (e.g. PeriodicBC). It does not apply constant boundary conditions (e.g. non-zero DirichletBC).

    source

    IncompressibleNavierStokes.diffusion_mat Method
    julia
    diffusion_mat(
     setup
-) -> SparseArrays.SparseMatrixCSC{Tv, Int64} where Tv

    Diffusion matrix.

    source

    IncompressibleNavierStokes.divergence_mat Method
    julia
    divergence_mat(
+) -> SparseArrays.SparseMatrixCSC{Tv, Int64} where Tv

    Diffusion matrix.

    source

    IncompressibleNavierStokes.divergence_mat Method
    julia
    divergence_mat(
     setup
-) -> SparseArrays.SparseMatrixCSC{Tv, Int64} where Tv

    Divergence matrix.

    source

    IncompressibleNavierStokes.laplacian_mat Method
    julia
    laplacian_mat(setup) -> Any

    Get matrix for the Laplacian operator (for the pressure-Poisson equation). This matrix takes scalar field inputs restricted to the actual degrees of freedom.

    source

    IncompressibleNavierStokes.pad_scalarfield_mat Method
    julia
    pad_scalarfield_mat(setup) -> Any

    Create matrix for padding inner scalar field with boundary volumes. This can be useful for algorithms that require vectors with degrees of freedom only, and not the ghost volumes. To go back, simply transpose the matrix.

    See also: pad_vectorfield_mat.

    source

    IncompressibleNavierStokes.pad_vectorfield_mat Method
    julia
    pad_vectorfield_mat(
+) -> SparseArrays.SparseMatrixCSC{Tv, Int64} where Tv

    Divergence matrix.

    source

    IncompressibleNavierStokes.laplacian_mat Method
    julia
    laplacian_mat(setup) -> Any

    Get matrix for the Laplacian operator (for the pressure-Poisson equation). This matrix takes scalar field inputs restricted to the actual degrees of freedom.

    source

    IncompressibleNavierStokes.pad_scalarfield_mat Method
    julia
    pad_scalarfield_mat(setup) -> Any

    Create matrix for padding inner scalar field with boundary volumes. This can be useful for algorithms that require vectors with degrees of freedom only, and not the ghost volumes. To go back, simply transpose the matrix.

    See also: pad_vectorfield_mat.

    source

    IncompressibleNavierStokes.pad_vectorfield_mat Method
    julia
    pad_vectorfield_mat(
     setup
-) -> SparseArrays.SparseMatrixCSC{Tv, Int64} where Tv

    Create matrix for padding inner vector field with boundary volumes, similar to pad_scalarfield_mat.

    source

    IncompressibleNavierStokes.pressuregradient_mat Method
    julia
    pressuregradient_mat(
+) -> SparseArrays.SparseMatrixCSC{Tv, Int64} where Tv

    Create matrix for padding inner vector field with boundary volumes, similar to pad_scalarfield_mat.

    source

    IncompressibleNavierStokes.pressuregradient_mat Method
    julia
    pressuregradient_mat(
     setup
-) -> SparseArrays.SparseMatrixCSC{Tv, Int64} where Tv

    Pressure gradient matrix.

    source

    IncompressibleNavierStokes.volume_mat Method
    julia
    volume_mat(setup) -> Any

    Volume-size matrix.

    source

    - +) -> SparseArrays.SparseMatrixCSC{Tv, Int64} where Tv

    Pressure gradient matrix.

    source

    IncompressibleNavierStokes.volume_mat Method
    julia
    volume_mat(setup) -> Any

    Volume-size matrix.

    source

    + \ No newline at end of file diff --git a/dev/manual/ns.html b/dev/manual/ns.html index d1bfd63bf..01dc5003d 100644 --- a/dev/manual/ns.html +++ b/dev/manual/ns.html @@ -9,9 +9,9 @@ - + - + @@ -24,7 +24,7 @@
    IncompressibleNavierStokes.jl

    Appearance

    Incompressible Navier-Stokes equations ​

    The incompressible Navier-Stokes equations describe conservation of mass and conservation of momentum, which can be written as a divergence-free constraint and an evolution equation:

    ∇⋅u=0,∂u∂t+∇⋅(uuT)=−∇p+ν∇2u+f,

    where Ω⊂Rd is the domain, d∈{2,3} is the spatial dimension, u=(u1,…,ud) is the velocity field, p is the pressure, ν is the kinematic viscosity, and f=(f1,…,fd) is the body force per unit of volume. The velocity, pressure, and body force are functions of the spatial coordinate x=(x1,…,xd) and time t. We assume that Ω is a rectangular domain.

    Integral form ​

    The integral form of the Navier-Stokes equations is used as starting point to develop a spatial discretization:

    1|O|∫∂Ou⋅ndΓ=0,ddt1|O|∫OudΩ=1|O|∫∂O(−uuT−pI+ν∇u)⋅ndΓ+1|O|∫OfdΩ,

    where O⊂Ω is an arbitrary control volume with boundary ∂O, normal n, surface element dΓ, and volume size |O|. We have divided by the control volume sizes in the integral form.

    Boundary conditions ​

    The boundary conditions on a part of the boundary Γ⊂∂Ω are one or more of the following:

    • Dirichlet: u=uBC on Γ for some uBC;

    • Neumann: ∇u⋅n=0 on Γ;

    • Periodic: u(x)=u(x+τ) and p(x)=p(x+τ) for x∈Γ, where Γ+τ⊂∂Ω is another part of the boundary and τ is a translation vector;

    • Stress free: σ⋅n=0 on Γ, where σ=(−pI+2νS).

    Pressure equation ​

    Taking the divergence of the momemtum equations yields a Poisson equation for the pressure:

    −∇2p=∇⋅(∇⋅(uuT))−∇⋅f

    In scalar notation, this becomes

    −∑α=1d∂2∂xα∂xαp=∑α=1d∑β=1d∂2∂xα∂xβ(uαuβ)−∑α=1d∂fα∂xα.

    Note the absence of time derivatives in the pressure equation. While the velocity field evolves in time, the pressure only changes such that the velocity stays divergence free.

    If there are no pressure boundary conditions, the pressure is only unique up to a constant. We set this constant to 1.

    Other quantities of interest ​

    Reynolds number ​

    The Reynolds number is the inverse of the viscosity: Re=1ν. It is the only flow parameter governing the incompressible Navier-Stokes equations.

    Kinetic energy ​

    The local and total kinetic energy are defined by k=12∥u∥22 and K=12∥u∥L2(Ω)2=∫ΩkdΩ.

    Vorticity ​

    The vorticity is defined as ω=∇×u.

    In 2D, it is a scalar field given by

    ω=−∂u1∂x2+∂u2∂x1.

    In 3D, it is a vector field given by

    ω=(−∂u2∂x3+∂u3∂x2−∂u3∂x1+∂u1∂x3−∂u1∂x2+∂u2∂x1).

    Note that the 2D vorticity is equal to the x3-component of the 3D vorticity.

    Stream function ​

    In 2D, the stream function ψ is a scalar field such that

    u1=∂ψ∂x2,u2=−∂ψ∂x1.

    It can be found by solving

    ∇2ψ=−ω.

    In 3D, the stream function is a vector field such that

    u=∇×ψ.

    It can be found by solving

    ∇2ψ=∇×ω.
    - + \ No newline at end of file diff --git a/dev/manual/operators.html b/dev/manual/operators.html index 453f3a1df..44cf6dcdb 100644 --- a/dev/manual/operators.html +++ b/dev/manual/operators.html @@ -9,11 +9,11 @@ - + - + - + @@ -23,13 +23,13 @@ -
    IncompressibleNavierStokes.jl

    Appearance

    Operators ​

    All discrete operators are built using KernelAbstractions.jl and Cartesian indices, similar to WaterLily.jl. This allows for dimension- and backend-agnostic code. See this blog post for how to write kernels. IncompressibleNavierStokes previously relied on assembling sparse operators to perform the same operations. While being very efficient and also compatible with CUDA (CUSPARSE), storing these matrices in memory is expensive for large 3D problems.

    IncompressibleNavierStokes.Offset Type
    julia
    struct Offset{D}

    Cartesian index unit vector in D = 2 or D = 3 dimensions. Calling Offset(D)(α) returns a Cartesian index with 1 in the dimension α and zeros elsewhere.

    See https://b-fg.github.io/2023/05/07/waterlily-on-gpu.html for writing kernel loops using Cartesian indices.

    Fields

    source

    IncompressibleNavierStokes.Dfield! Method
    julia
    Dfield!(d, G, p, setup; ϵ) -> Any

    Compute the D-field (in-place version).

    source

    IncompressibleNavierStokes.Dfield Method
    julia
    Dfield(p, setup; kwargs...) -> Any

    Compute the D-field [5] given by

    D=2|∇p|∇2p.

    Differentiable version.

    source

    IncompressibleNavierStokes.Qfield! Method
    julia
    Qfield!(Q, u, setup) -> Any

    Compute the Q-field (in-place version).

    source

    IncompressibleNavierStokes.Qfield Method
    julia
    Qfield(u, setup) -> Any

    Compute Q-field [6] given by

    Q=−12∑α,β∂uα∂xβ∂uβ∂xα.

    Differentiable version.

    source

    IncompressibleNavierStokes.applybodyforce! Method
    julia
    applybodyforce!(F, u, t, setup) -> Any

    Compute body force (in-place version). Add the result to F.

    source

    IncompressibleNavierStokes.applybodyforce Method
    julia
    applybodyforce(u, t, setup) -> Any

    Compute body force (differentiable version).

    source

    IncompressibleNavierStokes.applypressure! Method
    julia
    applypressure!(u, p, setup) -> Any

    Subtract pressure gradient (in-place version).

    source

    IncompressibleNavierStokes.applypressure Method
    julia
    applypressure(u, p, setup) -> Any

    Subtract pressure gradient (differentiable version).

    source

    IncompressibleNavierStokes.avg Method
    julia
    avg(ϕ, Δ, I, α) -> Any

    Average scalar field ϕ in the α-direction.

    source

    IncompressibleNavierStokes.convection! Method
    julia
    convection!(F, u, setup) -> Any

    Compute convective term (in-place version). Add the result to F.

    source

    IncompressibleNavierStokes.convection Method
    julia
    convection(u, setup) -> Any

    Compute convective term (differentiable version).

    source

    IncompressibleNavierStokes.convection_diffusion_temp! Method
    julia
    convection_diffusion_temp!(c, u, temp, setup) -> Any

    Compute convection-diffusion term for the temperature equation. (in-place version). Add result to c.

    source

    IncompressibleNavierStokes.convection_diffusion_temp Method
    julia
    convection_diffusion_temp(u, temp, setup) -> Any

    Compute convection-diffusion term for the temperature equation. (differentiable version).

    source

    IncompressibleNavierStokes.convectiondiffusion! Method
    julia
    convectiondiffusion!(F, u, setup) -> Any

    Compute convective and diffusive terms (in-place version). Add the result to F.

    source

    IncompressibleNavierStokes.diffusion! Method
    julia
    diffusion!(F, u, setup; use_viscosity) -> Any

    Compute diffusive term (in-place version). Add the result to F.

    Keyword arguments:

    • with_viscosity = true: Include viscosity in the operator.

    source

    IncompressibleNavierStokes.diffusion Method
    julia
    diffusion(u, setup; kwargs...) -> Any

    Compute diffusive term (differentiable version).

    source

    IncompressibleNavierStokes.dissipation! Method
    julia
    dissipation!(diss, diff, u, setup) -> Any

    Compute dissipation term for the temperature equation (in-place version). Add result to diss.

    source

    IncompressibleNavierStokes.dissipation Method
    julia
    dissipation(u, setup) -> Any

    Compute dissipation term for the temperature equation (differentiable version).

    source

    IncompressibleNavierStokes.dissipation_from_strain! Method
    julia
    dissipation_from_strain!(ϵ, u, setup) -> Any

    Compute dissipation term from strain-rate tensor (in-place version).

    source

    IncompressibleNavierStokes.dissipation_from_strain Method
    julia
    dissipation_from_strain(u, setup) -> Any

    Compute dissipation term 2ν⟨SijSij⟩ from strain-rate tensor (differentiable version).

    source

    IncompressibleNavierStokes.divergence! Method
    julia
    divergence!(div, u, setup) -> Any

    Compute divergence of velocity field (in-place version).

    source

    IncompressibleNavierStokes.divergence Method
    julia
    divergence(u, setup) -> Any

    Compute divergence of velocity field (differentiable version).

    source

    IncompressibleNavierStokes.divoftensor! Method
    julia
    divoftensor!(s, σ, setup) -> Any

    Compute divergence of a tensor with all components in the pressure points (in-place version). The stress tensors should be precomputed and stored in σ.

    source

    IncompressibleNavierStokes.eig2field! Method
    julia
    eig2field!(λ, u, setup) -> Any

    Compute the second eigenvalue of S2+R2 (in-place version).

    source

    IncompressibleNavierStokes.eig2field Method
    julia
    eig2field(u, setup) -> Any

    Compute the second eigenvalue of S2+R2, as proposed by Jeong and Hussain [6].

    Differentiable version.

    source

    IncompressibleNavierStokes.get_scale_numbers Method
    julia
    get_scale_numbers(
+    
    IncompressibleNavierStokes.jl
    Appearance
    Operators ​
    All discrete operators are built using KernelAbstractions.jl and Cartesian indices, similar to WaterLily.jl. This allows for dimension- and backend-agnostic code. See this blog post for how to write kernels. IncompressibleNavierStokes previously relied on assembling sparse operators to perform the same operations. While being very efficient and also compatible with CUDA (CUSPARSE), storing these matrices in memory is expensive for large 3D problems.
    IncompressibleNavierStokes.Offset Type
    julia
    struct Offset{D}
    Cartesian index unit vector in D = 2 or D = 3 dimensions. Calling Offset(D)(α) returns a Cartesian index with 1 in the dimension α and zeros elsewhere.
    See https://b-fg.github.io/2023/05/07/waterlily-on-gpu.html for writing kernel loops using Cartesian indices.
    Fields
    source
    IncompressibleNavierStokes.Dfield! Method
    julia
    Dfield!(d, G, p, setup; ϵ) -> Any
    Compute the D-field (in-place version).
    source
    IncompressibleNavierStokes.Dfield Method
    julia
    Dfield(p, setup; kwargs...) -> Any
    Compute the D-field [5] given by
    D=2|∇p|∇2p.
    Differentiable version.
    source
    IncompressibleNavierStokes.Qfield! Method
    julia
    Qfield!(Q, u, setup) -> Any
    Compute the Q-field (in-place version).
    source
    IncompressibleNavierStokes.Qfield Method
    julia
    Qfield(u, setup) -> Any
    Compute Q-field [6] given by
    Q=−12∑α,β∂uα∂xβ∂uβ∂xα.
    Differentiable version.
    source
    IncompressibleNavierStokes.applybodyforce! Method
    julia
    applybodyforce!(F, u, t, setup) -> Any
    Compute body force (in-place version). Add the result to F.
    source
    IncompressibleNavierStokes.applybodyforce Method
    julia
    applybodyforce(u, t, setup) -> Any
    Compute body force (differentiable version).
    source
    IncompressibleNavierStokes.applypressure! Method
    julia
    applypressure!(u, p, setup) -> Any
    Subtract pressure gradient (in-place version).
    source
    IncompressibleNavierStokes.applypressure Method
    julia
    applypressure(u, p, setup) -> Any
    Subtract pressure gradient (differentiable version).
    source
    IncompressibleNavierStokes.avg Method
    julia
    avg(ϕ, Δ, I, α) -> Any
    Average scalar field ϕ in the α-direction.
    source
    IncompressibleNavierStokes.convection! Method
    julia
    convection!(F, u, setup) -> Any
    Compute convective term (in-place version). Add the result to F.
    source
    IncompressibleNavierStokes.convection Method
    julia
    convection(u, setup) -> Any
    Compute convective term (differentiable version).
    source
    IncompressibleNavierStokes.convection_diffusion_temp! Method
    julia
    convection_diffusion_temp!(c, u, temp, setup) -> Any
    Compute convection-diffusion term for the temperature equation. (in-place version). Add result to c.
    source
    IncompressibleNavierStokes.convection_diffusion_temp Method
    julia
    convection_diffusion_temp(u, temp, setup) -> Any
    Compute convection-diffusion term for the temperature equation. (differentiable version).
    source
    IncompressibleNavierStokes.convectiondiffusion! Method
    julia
    convectiondiffusion!(F, u, setup) -> Any
    Compute convective and diffusive terms (in-place version). Add the result to F.
    source
    IncompressibleNavierStokes.diffusion! Method
    julia
    diffusion!(F, u, setup; use_viscosity) -> Any
    Compute diffusive term (in-place version). Add the result to F.
    Keyword arguments:
    • with_viscosity = true: Include viscosity in the operator.
    source
    IncompressibleNavierStokes.diffusion Method
    julia
    diffusion(u, setup; kwargs...) -> Any
    Compute diffusive term (differentiable version).
    source
    IncompressibleNavierStokes.dissipation! Method
    julia
    dissipation!(diss, diff, u, setup) -> Any
    Compute dissipation term for the temperature equation (in-place version). Add result to diss.
    source
    IncompressibleNavierStokes.dissipation Method
    julia
    dissipation(u, setup) -> Any
    Compute dissipation term for the temperature equation (differentiable version).
    source
    IncompressibleNavierStokes.dissipation_from_strain! Method
    julia
    dissipation_from_strain!(ϵ, u, setup) -> Any
    Compute dissipation term from strain-rate tensor (in-place version).
    source
    IncompressibleNavierStokes.dissipation_from_strain Method
    julia
    dissipation_from_strain(u, setup) -> Any
    Compute dissipation term 2ν⟨SijSij⟩ from strain-rate tensor (differentiable version).
    source
    IncompressibleNavierStokes.divergence! Method
    julia
    divergence!(div, u, setup) -> Any
    Compute divergence of velocity field (in-place version).
    source
    IncompressibleNavierStokes.divergence Method
    julia
    divergence(u, setup) -> Any
    Compute divergence of velocity field (differentiable version).
    source
    IncompressibleNavierStokes.divoftensor! Method
    julia
    divoftensor!(s, σ, setup) -> Any
    Compute divergence of a tensor with all components in the pressure points (in-place version). The stress tensors should be precomputed and stored in σ.
    source
    IncompressibleNavierStokes.eig2field! Method
    julia
    eig2field!(λ, u, setup) -> Any
    Compute the second eigenvalue of S2+R2 (in-place version).
    source
    IncompressibleNavierStokes.eig2field Method
    julia
    eig2field(u, setup) -> Any
    Compute the second eigenvalue of S2+R2, as proposed by Jeong and Hussain [6].
    Differentiable version.
    source
    IncompressibleNavierStokes.get_scale_numbers Method
    julia
    get_scale_numbers(
     u,
     setup
-) -> NamedTuple{(:uavg, :ϵ, :η, :λ, :Reλ, :L, :τ), <:Tuple{Any, Any, Any, Any, Any, Nothing, Nothing}}
    Get the following dimensional scale numbers [7]:
    • Velocity uavg=⟨uiui⟩1/2
    • Dissipation rate ϵ=2ν⟨SijSij⟩
    • Kolmolgorov length scale η=(ν3ϵ)1/4
    • Taylor length scale λ=(5νϵ)1/2uavg
    • Taylor-scale Reynolds number Reλ=λuavg3ν
    • Integral length scale L=3π2uavg2∫0∞E(k)kdk
    • Large-eddy turnover time τ=Luavg
    source
    IncompressibleNavierStokes.gravity! Method
    julia
    gravity!(F, temp, setup) -> Any
    Compute gravity term (in-place version). add the result to F.
    source
    IncompressibleNavierStokes.gravity Method
    julia
    gravity(temp, setup) -> Any
    Compute gravity term (differentiable version).
    source
    IncompressibleNavierStokes.interpolate_u_p! Method
    julia
    interpolate_u_p!(up, u, setup) -> Any
    Interpolate velocity to pressure points (in-place version).
    source
    IncompressibleNavierStokes.interpolate_u_p Method
    julia
    interpolate_u_p(u, setup) -> Any
    Interpolate velocity to pressure points (differentiable version).
    source
    IncompressibleNavierStokes.interpolate_ω_p! Method
    julia
    interpolate_ω_p!(ωp, ω, setup) -> Any
    Interpolate vorticity to pressure points (in-place version).
    source
    IncompressibleNavierStokes.interpolate_ω_p Method
    julia
    interpolate_ω_p(ω, setup) -> Any
    Interpolate vorticity to pressure points (differentiable version).
    source
    IncompressibleNavierStokes.kinetic_energy! Method
    julia
    kinetic_energy!(ke, u, setup; interpolate_first) -> Any
    Compute kinetic energy field (in-place version).
    source
    IncompressibleNavierStokes.kinetic_energy Method
    julia
    kinetic_energy(u, setup; kwargs...) -> Any
    Compute kinetic energy field k (in-place version). If interpolate_first is true, it is given by
    kI=18∑α(uI+hαα+uI−hαα)2.
    Otherwise, it is given by
    kI=14∑α((uI+hαα)2+(uI−hαα)2),
    as in [8].
    Differentiable version.
    source
    IncompressibleNavierStokes.laplacian! Method
    julia
    laplacian!(L, p, setup) -> Any
    Compute Laplacian of pressure field (in-place version).
    source
    IncompressibleNavierStokes.laplacian Method
    julia
    laplacian(p, setup) -> Any
    Compute Laplacian of pressure field (differentiable version).
    source
    IncompressibleNavierStokes.momentum! Method
    julia
    momentum!(F, u, temp, t, setup) -> Any
    Right hand side of momentum equations, excluding pressure gradient (in-place version).
    source
    IncompressibleNavierStokes.momentum Method
    julia
    momentum(u, temp, t, setup) -> Any
    Right hand side of momentum equations, excluding pressure gradient (differentiable version).
    source
    IncompressibleNavierStokes.pressuregradient! Method
    julia
    pressuregradient!(G, p, setup) -> Any
    Compute pressure gradient (in-place version).
    source
    IncompressibleNavierStokes.pressuregradient Method
    julia
    pressuregradient(p, setup) -> Any
    Compute pressure gradient (differentiable version).
    source
    IncompressibleNavierStokes.scalewithvolume! Method
    julia
    scalewithvolume!(p, setup) -> Any
    Scale scalar field with volume sizes (in-place version).
    source
    IncompressibleNavierStokes.scalewithvolume Method
    julia
    scalewithvolume(p, setup) -> Any
    Scale scalar field p with volume sizes (differentiable version).
    source
    IncompressibleNavierStokes.smagorinsky_closure Method
    julia
    smagorinsky_closure(
+) -> NamedTuple{(:uavg, :ϵ, :η, :λ, :Reλ, :L, :τ), <:Tuple{Any, Any, Any, Any, Any, Nothing, Nothing}}
    Get the following dimensional scale numbers [7]:
    • Velocity uavg=⟨uiui⟩1/2
    • Dissipation rate ϵ=2ν⟨SijSij⟩
    • Kolmolgorov length scale η=(ν3ϵ)1/4
    • Taylor length scale λ=(5νϵ)1/2uavg
    • Taylor-scale Reynolds number Reλ=λuavg3ν
    • Integral length scale L=3π2uavg2∫0∞E(k)kdk
    • Large-eddy turnover time τ=Luavg
    source
    IncompressibleNavierStokes.gravity! Method
    julia
    gravity!(F, temp, setup) -> Any
    Compute gravity term (in-place version). add the result to F.
    source
    IncompressibleNavierStokes.gravity Method
    julia
    gravity(temp, setup) -> Any
    Compute gravity term (differentiable version).
    source
    IncompressibleNavierStokes.interpolate_u_p! Method
    julia
    interpolate_u_p!(up, u, setup) -> Any
    Interpolate velocity to pressure points (in-place version).
    source
    IncompressibleNavierStokes.interpolate_u_p Method
    julia
    interpolate_u_p(u, setup) -> Any
    Interpolate velocity to pressure points (differentiable version).
    source
    IncompressibleNavierStokes.interpolate_ω_p! Method
    julia
    interpolate_ω_p!(ωp, ω, setup) -> Any
    Interpolate vorticity to pressure points (in-place version).
    source
    IncompressibleNavierStokes.interpolate_ω_p Method
    julia
    interpolate_ω_p(ω, setup) -> Any
    Interpolate vorticity to pressure points (differentiable version).
    source
    IncompressibleNavierStokes.kinetic_energy! Method
    julia
    kinetic_energy!(ke, u, setup; interpolate_first) -> Any
    Compute kinetic energy field (in-place version).
    source
    IncompressibleNavierStokes.kinetic_energy Method
    julia
    kinetic_energy(u, setup; kwargs...) -> Any
    Compute kinetic energy field k (in-place version). If interpolate_first is true, it is given by
    kI=18∑α(uI+hαα+uI−hαα)2.
    Otherwise, it is given by
    kI=14∑α((uI+hαα)2+(uI−hαα)2),
    as in [8].
    Differentiable version.
    source
    IncompressibleNavierStokes.laplacian! Method
    julia
    laplacian!(L, p, setup) -> Any
    Compute Laplacian of pressure field (in-place version).
    source
    IncompressibleNavierStokes.laplacian Method
    julia
    laplacian(p, setup) -> Any
    Compute Laplacian of pressure field (differentiable version).
    source
    IncompressibleNavierStokes.momentum! Method
    julia
    momentum!(F, u, temp, t, setup) -> Any
    Right hand side of momentum equations, excluding pressure gradient (in-place version).
    source
    IncompressibleNavierStokes.momentum Method
    julia
    momentum(u, temp, t, setup) -> Any
    Right hand side of momentum equations, excluding pressure gradient (differentiable version).
    source
    IncompressibleNavierStokes.pressuregradient! Method
    julia
    pressuregradient!(G, p, setup) -> Any
    Compute pressure gradient (in-place version).
    source
    IncompressibleNavierStokes.pressuregradient Method
    julia
    pressuregradient(p, setup) -> Any
    Compute pressure gradient (differentiable version).
    source
    IncompressibleNavierStokes.scalewithvolume! Method
    julia
    scalewithvolume!(p, setup) -> Any
    Scale scalar field with volume sizes (in-place version).
    source
    IncompressibleNavierStokes.scalewithvolume Method
    julia
    scalewithvolume(p, setup) -> Any
    Scale scalar field p with volume sizes (differentiable version).
    source
    IncompressibleNavierStokes.smagorinsky_closure Method
    julia
    smagorinsky_closure(
     setup
-) -> IncompressibleNavierStokes.var"#closure#188"
    Create Smagorinsky closure model m. The model is called as m(u, θ), where the Smagorinsky constant θ should be a scalar between 0 and 1 (for example θ = 0.1).
    source
    IncompressibleNavierStokes.smagtensor! Method
    julia
    smagtensor!(σ, u, θ, setup) -> Any
    Compute Smagorinsky stress tensors σ[I] (in-place version). The Smagorinsky constant θ should be a scalar between 0 and 1.
    source
    IncompressibleNavierStokes.total_kinetic_energy Method
    julia
    total_kinetic_energy(u, setup; kwargs...) -> Any
    Compute total kinetic energy. The velocity components are interpolated to the volume centers and squared.
    source
    IncompressibleNavierStokes.unit_cartesian_indices Method
    julia
    unit_cartesian_indices(D) -> Any
    Get tuple of all unit vectors as Cartesian indices.
    source
    IncompressibleNavierStokes.vorticity! Method
    julia
    vorticity!(ω, u, setup) -> Any
    Compute vorticity field (in-place version).
    source
    IncompressibleNavierStokes.vorticity Method
    julia
    vorticity(u, setup) -> Any
    Compute vorticity field (differentiable version).
    source
    IncompressibleNavierStokes.lastdimcontract Method
    julia
    lastdimcontract(a, b, setup) -> Any
    Compute c[I] = sum_i a[I, i] * b[I, i], where c[:, i] and b[:, i] are tensor fields (elements are SMatrixes), a[:, i] is a scalar field, and the i dimension is contracted (multiple channels).
    source
    IncompressibleNavierStokes.tensorbasis! Method
    julia
    tensorbasis!(B, V, u, setup) -> Tuple{Any, Any}
    Compute symmetry tensor basis B[1]-B[11] and invariants V[1]-V[5], as specified in [9] in equations (9) and (11). Note that B[1] corresponds to T0 in the paper, and V to I.
    source
    IncompressibleNavierStokes.tensorbasis Method
    julia
    tensorbasis(u, setup) -> Tuple{Any, Any}
    Compute symmetry tensor basis (differentiable version).
    source
    
-    
+) -> IncompressibleNavierStokes.var"#closure#188"

    Create Smagorinsky closure model m. The model is called as m(u, θ), where the Smagorinsky constant θ should be a scalar between 0 and 1 (for example θ = 0.1).

    source

    IncompressibleNavierStokes.smagtensor! Method
    julia
    smagtensor!(σ, u, θ, setup) -> Any

    Compute Smagorinsky stress tensors σ[I] (in-place version). The Smagorinsky constant θ should be a scalar between 0 and 1.

    source

    IncompressibleNavierStokes.total_kinetic_energy Method
    julia
    total_kinetic_energy(u, setup; kwargs...) -> Any

    Compute total kinetic energy. The velocity components are interpolated to the volume centers and squared.

    source

    IncompressibleNavierStokes.unit_cartesian_indices Method
    julia
    unit_cartesian_indices(D) -> Any

    Get tuple of all unit vectors as Cartesian indices.

    source

    IncompressibleNavierStokes.vorticity! Method
    julia
    vorticity!(ω, u, setup) -> Any

    Compute vorticity field (in-place version).

    source

    IncompressibleNavierStokes.vorticity Method
    julia
    vorticity(u, setup) -> Any

    Compute vorticity field (differentiable version).

    source

    IncompressibleNavierStokes.lastdimcontract Method
    julia
    lastdimcontract(a, b, setup) -> Any

    Compute c[I] = sum_i a[I, i] * b[I, i], where c[:, i] and b[:, i] are tensor fields (elements are SMatrixes), a[:, i] is a scalar field, and the i dimension is contracted (multiple channels).

    source

    IncompressibleNavierStokes.tensorbasis! Method
    julia
    tensorbasis!(B, V, u, setup) -> Tuple{Any, Any}

    Compute symmetry tensor basis B[1]-B[11] and invariants V[1]-V[5], as specified in [9] in equations (9) and (11). Note that B[1] corresponds to T0 in the paper, and V to I.

    source

    IncompressibleNavierStokes.tensorbasis Method
    julia
    tensorbasis(u, setup) -> Tuple{Any, Any}

    Compute symmetry tensor basis (differentiable version).

    source

    + \ No newline at end of file diff --git a/dev/manual/precision.html b/dev/manual/precision.html index 81f6b04ee..968e4ae0d 100644 --- a/dev/manual/precision.html +++ b/dev/manual/precision.html @@ -9,9 +9,9 @@ - + - + @@ -24,7 +24,7 @@
    IncompressibleNavierStokes.jl

    Appearance

    Floating point precision ​

    IncompressibleNavierStokes generates efficient code for different floating point precisions, such as

    • Double precision (Float64)

    • Single precision (Float32)

    • Half precision (Float16)

    To use single or half precision, all user input floats should be converted to the desired type. Mixing different precisions causes unnecessary conversions and may break the code.

    GPU precision

    For GPUs, single precision is preferred. CUDA.jls cu converts to single precision.

    Pressure solvers

    SparseArrays.jls sparse matrix factorizations only support double precision. psolver_direct only works for Float64. Consider using an iterative solver such as psolver_cg when using single or half precision.

    - + \ No newline at end of file diff --git a/dev/manual/pressure.html b/dev/manual/pressure.html index c2117e71f..9b60c49ed 100644 --- a/dev/manual/pressure.html +++ b/dev/manual/pressure.html @@ -9,11 +9,11 @@ - + - + - + @@ -25,21 +25,21 @@
    IncompressibleNavierStokes.jl

    Appearance

    Pressure solvers ​

    The discrete pressure Poisson equation

    Lp=WMF(u)

    enforces divergence freeness. There are multiple options for solving this system.

    IncompressibleNavierStokes.default_psolver Method
    julia
    default_psolver(
     setup
-) -> Union{IncompressibleNavierStokes.var"#psolve!#101"{Bool}, IncompressibleNavierStokes.var"#psolve!#124"}

    Get default Poisson solver from setup.

    source

    IncompressibleNavierStokes.poisson! Method
    julia
    poisson!(psolver, f) -> Any

    Solve the Poisson equation for the pressure (in-place version).

    source

    IncompressibleNavierStokes.poisson Method
    julia
    poisson(psolver, f) -> Any

    Solve the Poisson equation for the pressure with right hand side f at time t. For periodic and no-slip BC, the sum of f should be zero.

    Differentiable version.

    source

    IncompressibleNavierStokes.pressure! Method
    julia
    pressure!(p, u, temp, t, setup; psolver, F)

    Compute pressure from velocity field (in-place version).

    source

    IncompressibleNavierStokes.pressure Method
    julia
    pressure(u, temp, t, setup; psolver)

    Compute pressure from velocity field. This makes the pressure compatible with the velocity field, resulting in same order pressure as velocity.

    Differentiable version.

    source

    IncompressibleNavierStokes.project! Method
    julia
    project!(u, setup; psolver, p)

    Project velocity field onto divergence-free space (in-place version).

    source

    IncompressibleNavierStokes.project Method
    julia
    project(u, setup; psolver)

    Project velocity field onto divergence-free space (differentiable version).

    source

    IncompressibleNavierStokes.psolver_cg Method
    julia
    psolver_cg(
+) -> Union{IncompressibleNavierStokes.var"#psolve!#101"{Bool}, IncompressibleNavierStokes.var"#psolve!#124"}

    Get default Poisson solver from setup.

    source

    IncompressibleNavierStokes.poisson! Method
    julia
    poisson!(psolver, f) -> Any

    Solve the Poisson equation for the pressure (in-place version).

    source

    IncompressibleNavierStokes.poisson Method
    julia
    poisson(psolver, f) -> Any

    Solve the Poisson equation for the pressure with right hand side f at time t. For periodic and no-slip BC, the sum of f should be zero.

    Differentiable version.

    source

    IncompressibleNavierStokes.pressure! Method
    julia
    pressure!(p, u, temp, t, setup; psolver, F)

    Compute pressure from velocity field (in-place version).

    source

    IncompressibleNavierStokes.pressure Method
    julia
    pressure(u, temp, t, setup; psolver)

    Compute pressure from velocity field. This makes the pressure compatible with the velocity field, resulting in same order pressure as velocity.

    Differentiable version.

    source

    IncompressibleNavierStokes.project! Method
    julia
    project!(u, setup; psolver, p)

    Project velocity field onto divergence-free space (in-place version).

    source

    IncompressibleNavierStokes.project Method
    julia
    project(u, setup; psolver)

    Project velocity field onto divergence-free space (differentiable version).

    source

    IncompressibleNavierStokes.psolver_cg Method
    julia
    psolver_cg(
     setup;
     abstol,
     reltol,
     maxiter,
     preconditioner
-) -> IncompressibleNavierStokes.var"#psolve!#111"{_A, _B, _C, IncompressibleNavierStokes.var"#laplace_diag#109"{ndrange, workgroupsize}} where {_A, _B, _C, ndrange, workgroupsize}

    Conjugate gradients iterative Poisson solver.

    source

    IncompressibleNavierStokes.psolver_cg_matrix Method
    julia
    psolver_cg_matrix(
+) -> IncompressibleNavierStokes.var"#psolve!#111"{_A, _B, _C, IncompressibleNavierStokes.var"#laplace_diag#109"{ndrange, workgroupsize}} where {_A, _B, _C, ndrange, workgroupsize}

    Conjugate gradients iterative Poisson solver.

    source

    IncompressibleNavierStokes.psolver_cg_matrix Method
    julia
    psolver_cg_matrix(
     setup;
     kwargs...
-) -> IncompressibleNavierStokes.var"#psolve!#105"{Base.Pairs{Symbol, Union{}, Tuple{}, @NamedTuple{}}}

    Conjugate gradients iterative Poisson solver. The kwargs are passed to the cg! function from IterativeSolvers.jl.

    source

    IncompressibleNavierStokes.psolver_direct Method
    julia
    psolver_direct(
+) -> IncompressibleNavierStokes.var"#psolve!#105"{Base.Pairs{Symbol, Union{}, Tuple{}, @NamedTuple{}}}

    Conjugate gradients iterative Poisson solver. The kwargs are passed to the cg! function from IterativeSolvers.jl.

    source

    IncompressibleNavierStokes.psolver_direct Method
    julia
    psolver_direct(
     setup
-) -> IncompressibleNavierStokes.var"#psolve!#101"{Bool}

    Create direct Poisson solver using an appropriate matrix decomposition.

    source

    IncompressibleNavierStokes.psolver_spectral Method
    julia
    psolver_spectral(
+) -> IncompressibleNavierStokes.var"#psolve!#101"{Bool}

    Create direct Poisson solver using an appropriate matrix decomposition.

    source

    IncompressibleNavierStokes.psolver_spectral Method
    julia
    psolver_spectral(
     setup
-) -> IncompressibleNavierStokes.var"#psolve!#124"

    Create spectral Poisson solver from setup.

    source

    - +) -> IncompressibleNavierStokes.var"#psolve!#124"

    Create spectral Poisson solver from setup.

    source

    + \ No newline at end of file diff --git a/dev/manual/sciml.html b/dev/manual/sciml.html index 355d43018..4293d3f68 100644 --- a/dev/manual/sciml.html +++ b/dev/manual/sciml.html @@ -9,11 +9,11 @@ - + - + - + @@ -34,48 +34,48 @@ problem = ODEProblem(f, u0, tspan) #SciMLBase.ODEProblem sol = solve(problem, Tsit5(), reltol = 1e-8, abstol = 1e-8) # sol: SciMLBase.ODESolution
    retcode: Success
 Interpolation: specialized 4th order "free" interpolation
-t: 231-element Vector{Float64}:
+t: 204-element Vector{Float64}:
  0.0
- 0.0016659277128752055
- 0.002432989100957199
- 0.0036642837413415376
- 0.004709119986944817
- 0.005918078594056992
- 0.007106129763889296
- 0.00836604328705504
- 0.00964424423128464
- 0.010963315612198047
+ 0.0017630725463301349
+ 0.002578765391583634
+ 0.0038900689003174543
+ 0.0050004385667972275
+ 0.006282968599569631
+ 0.00753879858193828
+ 0.008865534083334846
+ 0.01020691087682567
+ 0.011586923053397053
  ⋮
- 0.901224250082742
- 0.9130080474847379
- 0.9249889455158832
- 0.9371708873148851
- 0.9495567173255886
- 0.9621483766861515
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+ 0.9061469742584687
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+ 0.9327544152402584
+ 0.9463937370590901
+ 0.9602625548035638
+ 0.9743640183560942
+ 0.9887005249941152
  1.0
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+u: 204-element Vector{Array{Float64, 3}}:
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  ⋮
- [-2.6008269440350635 -2.6707555983699267 … -2.6008269440350635 -2.6707555983699267; -1.7530707144997295 0.09689754026634612 … 0.08931652432247436 -1.5878474701844734; … ; -2.6008269440350635 0.10720478554028837 … 0.10270212910901891 -2.6707555983699267; -1.7530707144997295 -1.5878474701844734 … -1.7530707144997295 -1.5878474701844734;;; -0.33445088457939787 0.7427051143709156 … -0.33445088457939787 0.7427051143709156; -0.3997760803661724 0.49976125856316267 … 0.48945401328925586 1.0678539731154677; … ; -0.33445088457939787 0.49375215219778495 … 0.48034789830989794 0.7427051143709156; -0.3997760803661724 1.0678539731154677 … -0.3997760803661724 1.0678539731154677]
- [-2.601725136184899 -2.670873685384534 … -2.601725136184899 -2.670873685384534; -1.7563563398627091 0.10432541979422275 … 0.09676710645672201 -1.5905130283970155; … ; -2.601725136184899 0.11653309235413584 … 0.11187293004895257 -2.670873685384534; -1.7563563398627091 -1.5905130283970155 … -1.7563563398627091 -1.5905130283970155;;; -0.3522441846097827 0.7271704846904127 … -0.3522441846097827 0.7271704846904127; -0.41329173959034654 0.49289307311066627 … 0.48068540055078907 1.056633316058722; … ; -0.3522441846097827 0.48307344888160403 … 0.46792424402653476 0.7271704846904127; -0.41329173959034654 1.056633316058722 … -0.41329173959034654 1.056633316058722]
- [-2.6025523523428413 -2.67071744709628 … -2.6025523523428413 -2.67071744709628; -1.7595072686400752 0.11170467403179123 … 0.10398180790626674 -1.592875884389524; … ; -2.6025523523428413 0.12573523533040898 … 0.12069919884062454 -2.67071744709628; -1.7595072686400752 -1.592875884389524 … -1.7595072686400752 -1.592875884389524;;; -0.3716771235605255 0.7097434315940944 … -0.3716771235605255 0.7097434315940944; -0.42842794202681916 0.4845376104979045 … 0.470507049199323 1.04360335395341; … ; -0.3716771235605255 0.4708320861058763 … 0.4540567364223937 0.7097434315940944; -0.42842794202681916 1.04360335395341 … -0.42842794202681916 1.04360335395341]
- [-2.6033919710125044 -2.6703849125869374 … -2.6033919710125044 -2.6703849125869374; -1.7625884932762157 0.11894958866689538 … 0.1108737142172876 -1.595010124997473; … ; -2.6033919710125044 0.1346989359254884 … 0.12906953547026517 -2.6703849125869374; -1.7625884932762157 -1.595010124997473 … -1.7625884932762157 -1.595010124997473;;; -0.3926613103435367 0.6904669028807235 … -0.3926613103435367 0.6904669028807235; -0.44514969973593976 0.4746627267546408 … 0.45891337949608374 1.0287556320125302; … ; -0.3926613103435367 0.4570355653834647 … 0.43877942685883076 0.6904669028807235; -0.44514969973593976 1.0287556320125302 … -0.44514969973593976 1.0287556320125302]
- [-2.6043280594687412 -2.6699787184933967 … -2.6043280594687412 -2.6699787184933967; -1.7656684357322596 0.12596507497113008 … 0.11734810375345404 -1.5969962068782388; … ; -2.6043280594687412 0.14330182663680066 … 0.13686518330313233 -2.6699787184933967; -1.7656684357322596 -1.5969962068782388 … -1.7656684357322596 -1.5969962068782388;;; -0.4150660872718007 0.6694280502383273 … -0.4150660872718007 0.6694280502383273; -0.4633875966731134 0.4632574290511748 … 0.44592067738554064 1.0121162640308192; … ; -0.4150660872718007 0.44171996769365324 … 0.4221548844285295 0.6694280502383273; -0.4633875966731134 1.0121162640308192 … -0.4633875966731134 1.0121162640308192]
- [-2.60544291809869 -2.669603929032186 … -2.60544291809869 -2.669603929032186; -1.7688174940989976 0.1326468336290909 … 0.12330344096559884 -1.5989199688668567; … ; -2.60544291809869 0.15141245391774916 … 0.14396194354418868 -2.669603929032186; -1.7688174940989976 -1.5989199688668567 … -1.7688174940989976 -1.5989199688668567;;; -0.4387165340878582 0.6467626475342549 … -0.4387165340878582 0.6467626475342549; -0.48303489816549916 0.4503361141495976 … 0.4315704938609758 0.9937508037134614; … ; -0.4387165340878582 0.42495415887972643 … 0.4042773366315602 0.6467626475342549; -0.48303489816549916 0.9937508037134614 … -0.48303489816549916 0.9937508037134614]
- [-2.6068139332770977 -2.6693650234422095 … -2.6068139332770977 -2.6693650234422095; -1.77210571029155 0.13888179342208257 … 0.12863269582615977 -1.6008706855176826; … ; -2.6068139332770977 0.15889167346612212 … 0.15023250924374582 -2.6693650234422095; -1.77210571029155 -1.6008706855176826 … -1.77210571029155 -1.6008706855176826;;; -0.463391491674076 0.6226597041634463 … -0.463391491674076 0.6226597041634463; -0.5039442703140851 0.4359437228387498 … 0.4159338427947474 0.973769704494905; … ; -0.463391491674076 0.40684472968608854 … 0.385276367047268 0.6226597041634463; -0.5039442703140851 0.973769704494905 … -0.5039442703140851 0.973769704494905]
- [-2.608510095624186 -2.6693620397707933 … -2.608510095624186 -2.6693620397707933; -1.7756002089136793 0.14455026801749912 … 0.13322622135795462 -1.6029384976802836; … ; -2.608510095624186 0.1655961649508638 … 0.15555064935246116 -2.6693620397707933; -1.7756002089136793 -1.6029384976802836 … -1.7756002089136793 -1.6029384976802836;;; -0.4888266014764763 0.5973613484526877 … -0.4888266014764763 0.5973613484526877; -0.5259282049009588 0.4201590976985429 … 0.3991132007652158 0.952330459812817; … ; -0.4888266014764763 0.3875380448857541 … 0.365317344444688 0.5973613484526877; -0.5259282049009588 0.952330459812817 … -0.5259282049009588 0.952330459812817]
- [-2.610391429817098 -2.669642952553338 … -2.610391429817098 -2.669642952553338; -1.7790232145791847 0.14912668082348163 … 0.1366855419132724 -1.605003300278285; … ; -2.610391429817098 0.17091984605417793 … 0.15947240646934602 -2.669642952553338; -1.7790232145791847 -1.605003300278285 … -1.7790232145791847 -1.605003300278285;;; -0.512459536652462 0.5734647082450985 … -0.512459536652462 0.5734647082450985; -0.5467494943080825 0.4046273175965032 … 0.3828341523658447 0.9316537922498789; … ; -0.512459536652462 0.3690211337021703 … 0.3464251477003929 0.5734647082450985; -0.5467494943080825 0.9316537922498789 … -0.5467494943080825 0.9316537922498789]

    Alternatively, it is also possible to use an in-place formulation

    julia
    f!(du,u,p,t) = right_hand_side!(du, u, Ref([setup, psolver]), t)
+ [3.63658584761423 3.87080429663847 … 3.63658584761423 3.87080429663847; 2.6792423852120266 -0.1225160105596109 … -0.10545067559959928 2.802014516816849; … ; 3.63658584761423 -0.12912223894278943 … -0.11638439519314522 3.87080429663847; 2.6792423852120266 2.802014516816849 … 2.6792423852120266 2.802014516816849;;; -3.7994309843333527 -4.208021775307386 … -3.7994309843333527 -4.208021775307386; -3.5271355479570685 0.0368409287211739 … 0.04344715710435443 -4.001965753158543; … ; -3.7994309843333527 0.014947618503139795 … 0.018108013501957024 -4.208021775307386; -3.5271355479570685 -4.001965753158543 … -3.5271355479570685 -4.001965753158543]
+ [3.6422895086948026 3.8754147141679445 … 3.6422895086948026 3.8754147141679445; 2.6861592819208227 -0.1102895726892691 … -0.09162669561786173 2.807686775420112; … ; 3.6422895086948026 -0.1177386254998089 … -0.10346473297439138 3.8754147141679445; 2.6861592819208227 2.807686775420112 … 2.6861592819208227 2.807686775420112;;; -3.7931619925313242 -4.202317958546451 … -3.7931619925313242 -4.202317958546451; -3.520066837125464 0.03622302471012847 … 0.04367207752067032 -3.995334854799291; … ; -3.7931619925313242 0.013948554503623277 … 0.01797602011337264 -4.202317958546451; -3.520066837125464 -3.995334854799291 … -3.520066837125464 -3.995334854799291]
+ [3.6479937698880462 3.880149347154071 … 3.6479937698880462 3.880149347154071; 2.6930364840785233 -0.09783809929702272 … -0.07776985332212666 2.813469779247899; … ; 3.6479937698880462 -0.10613939385057908 … -0.09048860694780399 3.880149347154071; 2.6930364840785233 2.813469779247899 … 2.6930364840785233 2.813469779247899;;; -3.787054128421903 -4.19668244159331 … -3.787054128421903 -4.19668244159331; -3.5132283960223627 0.03581805063493754 … 0.04411934518849587 -3.988821699344505; … ; -3.787054128421903 0.013198507566411128 … 0.01809997695775903 -4.19668244159331; -3.5132283960223627 -3.988821699344505 … -3.5132283960223627 -3.988821699344505]
+ [3.653648807156927 3.884965007263431 … 3.653648807156927 3.884965007263431; 2.699812752232953 -0.0852337491691666 … -0.0639660324794772 2.819309423233572; … ; 3.653648807156927 -0.09438466632096755 … -0.07752884716293337 3.884965007263431; 2.699812752232953 2.819309423233572 … 2.699812752232953 2.819309423233572;;; -3.781138941703361 -4.191145491275563 … -3.781138941703361 -4.191145491275563; -3.506658733142349 0.03562880939265571 … 0.044779726544458685 -3.9824637292406186; … ; -3.781138941703361 0.012699036170763192 … 0.01847153724798791 -4.191145491275563; -3.506658733142349 -3.9824637292406186 … -3.506658733142349 -3.9824637292406186]
+ [3.659205979536996 3.889816751078532 … 3.659205979536996 3.889816751078532; 2.7064289741897443 -0.07255414846164322 … -0.05030101815559454 2.825149688747564; … ; 3.659205979536996 -0.08253980267461902 … -0.06465938047302927 3.889816751078532; 2.7064289741897443 2.825149688747564 … 2.7064289741897443 2.825149688747564;;; -3.775446647182436 -4.185737271493904 … -3.775446647182436 -4.185737271493904; -3.5003941254461775 0.03565465237106255 … 0.04564030658404039 -3.976298023468234; … ; -3.775446647182436 0.012448758810299665 … 0.019078978590910903 -4.185737271493904; -3.5003941254461775 -3.976298023468234 … -3.5003941254461775 -3.976298023468234]
+ [3.664618686922103 3.8946587976801346 … 3.664618686922103 3.8946587976801346; 2.7128292402547007 -0.05988085958027444 … -0.0368592383448486 2.8309339928553903; … ; 3.664618686922103 -0.07067439363603704 … -0.05195426801286364 3.8946587976801346; 2.7128292402547007 2.8309339928553903 … 2.7128292402547007 2.8309339928553903;;; -3.7700058000739514 -4.180487610093032 … -3.7700058000739514 -4.180487610093032; -3.4944680861051314 0.0358913760423405 … 0.046684910098105145 -3.97036081328936; … ; -3.7700058000739514 0.012443154773298683 … 0.019907432510493542 -4.180487610093032; -3.4944680861051314 -3.97036081328936 … -3.4944680861051314 -3.97036081328936]
+ [3.669843653298168 3.8994459366486214 … 3.669843653298168 3.8994459366486214; 2.7189623589035268 -0.04729660729185216 … -0.023721384830430667 2.836607060649013; … ; 3.669843653298168 -0.05886002218388251 … -0.03948559259932328 3.8994459366486214; 2.7189623589035268 2.836607060649013 … 2.7189623589035268 2.836607060649013;;; -3.7648424333013457 -4.175425182695787 … -3.7648424333013457 -4.175425182695787; -3.4889102867335025 0.03633131410140703 … 0.04789472899343934 -3.964686349775314; … ; -3.7648424333013457 0.012674494032257701 … 0.020939295360611808 -4.175425182695787; -3.4889102867335025 -3.964686349775314 … -3.4889102867335025 -3.964686349775314]
+ [3.674841850453165 3.904134731547883 … 3.674841850453165 3.904134731547883; 2.724782882886711 -0.03488300018573105 … -0.010962759787617392 2.8421164708795263; … ; 3.674841850453165 -0.04716839537064069 … -0.027321933899741135 3.904134731547883; 2.724782882886711 2.8421164708795263 … 2.724782882886711 2.8421164708795263;;; -3.759979480483631 -4.170576881038018 … -3.759979480483631 -4.170576881038018; -3.4837458594845088 0.036963640271927826 … 0.049249035456839486 -3.959306024454923; … ; -3.759979480483631 0.01313197193049889 … 0.022154757023338023 -4.170576881038018; -3.4837458594845088 -3.959306024454923 … -3.4837458594845088 -3.959306024454923]
+ [3.6785458556973847 3.9076829270352 … 3.6785458556973847 3.9076829270352; 2.7290632367754313 -0.02540851001379519 … -0.0013580255384598883 2.8462520790618577; … ; 3.6785458556973847 -0.0382162612055059 … -0.01812500342619521 3.9076829270352; 2.7290632367754313 2.8462520790618577 … 2.7290632367754313 2.8462520790618577;;; -3.7564219120903566 -4.1669743965658705 … -3.7564219120903566 -4.1669743965658705; -3.4800187689780997 0.037579010665183425 … 0.05038676185689626 -3.9553476091080677; … ; -3.7564219120903566 0.013634784336917961 … 0.023211947861768473 -4.1669743965658705; -3.4800187689780997 -3.9553476091080677 … -3.4800187689780997 -3.9553476091080677]

    Alternatively, it is also possible to use an in-place formulation

    julia
    f!(du,u,p,t) = right_hand_side!(du, u, Ref([setup, psolver]), t)
 u = similar(u0)
 du = similar(u0)
 p = nothing
@@ -83,8 +83,8 @@
 f!(du,u,p,t)

    that is usually faster than the out-of-place formulation.

    You can look here for more information on how to use the SciML solvers and all the options available.

    API ​

    IncompressibleNavierStokes.create_right_hand_side Method
    julia
    create_right_hand_side(
     setup,
     psolver
-) -> IncompressibleNavierStokes.var"#right_hand_side#343"
    create_right_hand_side(setup, psolver)

    Creates a function that computes the right-hand side of the Navier-Stokes equations for a given setup and pressure solver.

    Arguments

    • setup: The simulation setup containing grid and boundary conditions.

    • psolver: The pressure solver to be used.

    Returns

    A function that computes the right-hand side of the Navier-Stokes equations.

    source

    IncompressibleNavierStokes.right_hand_side! Method
    julia
    right_hand_side!(dudt, u, params_ref, t)
    right_hand_side!(dudt, u, params_ref, t)

    Computes the right-hand side of the Navier-Stokes equations in-place.

    Arguments

    • dudt: The array to store the computed right-hand side.

    • u: The current velocity field.

    • params_ref: A reference to the parameters containing the setup and pressure solver.

    • t: The current time.

    Returns

    Nothing. The result is stored in dudt.

    source

    - +) -> IncompressibleNavierStokes.var"#right_hand_side#343"
    create_right_hand_side(setup, psolver)

    Creates a function that computes the right-hand side of the Navier-Stokes equations for a given setup and pressure solver.

    Arguments

    • setup: The simulation setup containing grid and boundary conditions.

    • psolver: The pressure solver to be used.

    Returns

    A function that computes the right-hand side of the Navier-Stokes equations.

    source

    IncompressibleNavierStokes.right_hand_side! Method
    julia
    right_hand_side!(dudt, u, params_ref, t)
    right_hand_side!(dudt, u, params_ref, t)

    Computes the right-hand side of the Navier-Stokes equations in-place.

    Arguments

    • dudt: The array to store the computed right-hand side.

    • u: The current velocity field.

    • params_ref: A reference to the parameters containing the setup and pressure solver.

    • t: The current time.

    Returns

    Nothing. The result is stored in dudt.

    source

    + \ No newline at end of file diff --git a/dev/manual/setup.html b/dev/manual/setup.html index 7fe3fb556..7af6d504a 100644 --- a/dev/manual/setup.html +++ b/dev/manual/setup.html @@ -9,11 +9,11 @@ - + - + - + @@ -23,13 +23,13 @@ -
    IncompressibleNavierStokes.jl

    Appearance

    Problem setup ​

    Boundary conditions ​

    Each boundary has exactly one type of boundary conditions. For periodic boundary conditions, the opposite boundary must also be periodic.

    IncompressibleNavierStokes.AbstractBC Type
    julia
    abstract type AbstractBC

    Boundary condition for one side of the domain.

    Fields

    source

    IncompressibleNavierStokes.DirichletBC Type
    julia
    struct DirichletBC{U} <: IncompressibleNavierStokes.AbstractBC

    Dirichlet boundary conditions for the velocity, where u[1] = (x..., t) -> u1_BC up to u[d] = (x..., t) -> ud_BC, where d is the dimension.

    When u is nothing, then the boundary conditions are no slip boundary conditions, where all velocity components are zero.

    Fields

    • u: Boundary condition

    source

    IncompressibleNavierStokes.PeriodicBC Type
    julia
    struct PeriodicBC <: IncompressibleNavierStokes.AbstractBC

    Periodic boundary conditions. Must be periodic on both sides.

    Fields

    source

    IncompressibleNavierStokes.PressureBC Type
    julia
    struct PressureBC <: IncompressibleNavierStokes.AbstractBC

    Pressure boundary conditions. The pressure is prescribed on the boundary (usually an "outlet"). The velocity has zero Neumann conditions.

    Note: Currently, the pressure is prescribed with the constant value of zero on the entire boundary.

    Fields

    source

    IncompressibleNavierStokes.SymmetricBC Type
    julia
    struct SymmetricBC <: IncompressibleNavierStokes.AbstractBC

    Symmetric boundary conditions. The parallel velocity and pressure is the same at each side of the boundary. The normal velocity is zero.

    Fields

    source

    IncompressibleNavierStokes.apply_bc_p! Method
    julia
    apply_bc_p!(p, t, setup; kwargs...) -> Any

    Apply pressure boundary conditions (in-place version).

    source

    IncompressibleNavierStokes.apply_bc_p Method
    julia
    apply_bc_p(p, t, setup; kwargs...) -> Any

    Apply pressure boundary conditions (differentiable version).

    source

    IncompressibleNavierStokes.apply_bc_temp! Method
    julia
    apply_bc_temp!(temp, t, setup; kwargs...) -> Any

    Apply temperature boundary conditions (in-place version).

    source

    IncompressibleNavierStokes.apply_bc_temp Method
    julia
    apply_bc_temp(temp, t, setup; kwargs...) -> Any

    Apply temperature boundary conditions (differentiable version).

    source

    IncompressibleNavierStokes.apply_bc_u! Method
    julia
    apply_bc_u!(u, t, setup; kwargs...) -> Any

    Apply velocity boundary conditions (in-place version).

    source

    IncompressibleNavierStokes.apply_bc_u Method
    julia
    apply_bc_u(u, t, setup; kwargs...) -> Any

    Apply velocity boundary conditions (differentiable version).

    source

    IncompressibleNavierStokes.boundary Method
    julia
    boundary(β, N, I, isright) -> Any

    Get boundary indices of boundary layer normal to β. The CartesianIndices given by I should contain those of the inner DOFs, typically Ip or Iu[α]. The boundary layer is then just outside those.

    source

    IncompressibleNavierStokes.offset_p Function
    julia
    offset_p(bc, isnormal, isright)

    Number of non-DOF pressure components at boundary.

    source

    IncompressibleNavierStokes.offset_u Function
    julia
    offset_u(bc, isnormal, isright)

    Number of non-DOF velocity components at boundary. If isnormal, then the velocity is normal to the boundary, else parallel. If isright, it is at the end/right/rear/top boundary, otherwise beginning.

    source

    Grid ​

    IncompressibleNavierStokes.Dimension Type
    julia
    struct Dimension{N}

    Represent an N-dimensional space. Returns N when called.

    julia
    julia> d = Dimension(3)
+    
    IncompressibleNavierStokes.jl
    Appearance
    Problem setup ​
    Boundary conditions ​
    Each boundary has exactly one type of boundary conditions. For periodic boundary conditions, the opposite boundary must also be periodic.
    IncompressibleNavierStokes.AbstractBC Type
    julia
    abstract type AbstractBC
    Boundary condition for one side of the domain.
    Fields
    source
    IncompressibleNavierStokes.DirichletBC Type
    julia
    struct DirichletBC{U} <: IncompressibleNavierStokes.AbstractBC
    Dirichlet boundary conditions for the velocity, where u[1] = (x..., t) -> u1_BC up to u[d] = (x..., t) -> ud_BC, where d is the dimension.
    When u is nothing, then the boundary conditions are no slip boundary conditions, where all velocity components are zero.
    Fields
    • u: Boundary condition
    source
    IncompressibleNavierStokes.PeriodicBC Type
    julia
    struct PeriodicBC <: IncompressibleNavierStokes.AbstractBC
    Periodic boundary conditions. Must be periodic on both sides.
    Fields
    source
    IncompressibleNavierStokes.PressureBC Type
    julia
    struct PressureBC <: IncompressibleNavierStokes.AbstractBC
    Pressure boundary conditions. The pressure is prescribed on the boundary (usually an "outlet"). The velocity has zero Neumann conditions.
    Note: Currently, the pressure is prescribed with the constant value of zero on the entire boundary.
    Fields
    source
    IncompressibleNavierStokes.SymmetricBC Type
    julia
    struct SymmetricBC <: IncompressibleNavierStokes.AbstractBC
    Symmetric boundary conditions. The parallel velocity and pressure is the same at each side of the boundary. The normal velocity is zero.
    Fields
    source
    IncompressibleNavierStokes.apply_bc_p! Method
    julia
    apply_bc_p!(p, t, setup; kwargs...) -> Any
    Apply pressure boundary conditions (in-place version).
    source
    IncompressibleNavierStokes.apply_bc_p Method
    julia
    apply_bc_p(p, t, setup; kwargs...) -> Any
    Apply pressure boundary conditions (differentiable version).
    source
    IncompressibleNavierStokes.apply_bc_temp! Method
    julia
    apply_bc_temp!(temp, t, setup; kwargs...) -> Any
    Apply temperature boundary conditions (in-place version).
    source
    IncompressibleNavierStokes.apply_bc_temp Method
    julia
    apply_bc_temp(temp, t, setup; kwargs...) -> Any
    Apply temperature boundary conditions (differentiable version).
    source
    IncompressibleNavierStokes.apply_bc_u! Method
    julia
    apply_bc_u!(u, t, setup; kwargs...) -> Any
    Apply velocity boundary conditions (in-place version).
    source
    IncompressibleNavierStokes.apply_bc_u Method
    julia
    apply_bc_u(u, t, setup; kwargs...) -> Any
    Apply velocity boundary conditions (differentiable version).
    source
    IncompressibleNavierStokes.boundary Method
    julia
    boundary(β, N, I, isright) -> Any
    Get boundary indices of boundary layer normal to β. The CartesianIndices given by I should contain those of the inner DOFs, typically Ip or Iu[α]. The boundary layer is then just outside those.
    source
    IncompressibleNavierStokes.offset_p Function
    julia
    offset_p(bc, isnormal, isright)
    Number of non-DOF pressure components at boundary.
    source
    IncompressibleNavierStokes.offset_u Function
    julia
    offset_u(bc, isnormal, isright)
    Number of non-DOF velocity components at boundary. If isnormal, then the velocity is normal to the boundary, else parallel. If isright, it is at the end/right/rear/top boundary, otherwise beginning.
    source
    Grid ​
    IncompressibleNavierStokes.Dimension Type
    julia
    struct Dimension{N}
    Represent an N-dimensional space. Returns N when called.
    julia
    julia> d = Dimension(3)
 Dimension{3}()
 
 julia> d()
-3
    Fields
    source
    IncompressibleNavierStokes.Grid Method
    julia
    Grid(; x, boundary_conditions, backend)
    Create useful quantities for Cartesian box mesh ``x[1] \times \dots \times x[d]with boundary conditionsboundary_conditions. Return a named tuple ([α]` denotes a tuple index) with the following fields:
    • N[α]: Number of finite volumes in direction β, including ghost volumes
    • Nu[α][β]: Number of u[α] velocity DOFs in direction β
    • Np[α]: Number of pressure DOFs in direction α
    • Iu[α]: Cartesian index range of u[α] velocity DOFs
    • Ip: Cartesian index range of pressure DOFs
    • xlims[α]: Tuple containing the limits of the physical domain (not grid) in the direction α
    • x[α]: α-coordinates of all volume boundaries, including the left point of the first ghost volume
    • xu[α][β]: β-coordinates of u[α] velocity points
    • xp[α]: α-coordinates of pressure points
    • Δ[α]: All volume widths in direction α
    • Δu[α]: Distance between pressure points in direction α
    • A[α][β]: Interpolation weights from α-face centers xI to xI±hβ
    Note that the memory footprint of the redundant 1D-arrays above is negligible compared to the memory footprint of the 2D/3D-fields used in the code.
    source
    IncompressibleNavierStokes.cosine_grid Method
    julia
    cosine_grid(a, b, N) -> Any
    Create a nonuniform grid of N + 1 points from a to b using a cosine profile, i.e.
    xi=a+12(1−cos⁡(πin))(b−a),i=0,…,N
    See also stretched_grid.
    source
    IncompressibleNavierStokes.max_size Method
    julia
    max_size(grid) -> Any
    Get size of the largest grid element.
    source
    IncompressibleNavierStokes.stretched_grid Function
    julia
    stretched_grid(a, b, N) -> Any
-stretched_grid(a, b, N, s) -> Any
    Create a nonuniform grid of N + 1 points from a to b with a stretch factor of s. If s = 1, return a uniform spacing from a to b. Otherwise, return a vector x∈RN+1 such that xn=a+∑i=1nsi−1h for n=0,…,N. Setting xN=b then gives h=(b−a)1−s1−sN, resulting in
    xn=a+(b−a)1−sn1−sN,n=0,…,N.
    Note that stretched_grid(a, b, N, s)[n] corresponds to xn−1.
    See also cosine_grid.
    source
    IncompressibleNavierStokes.tanh_grid Function
    julia
    tanh_grid(a, b, N) -> Any
-tanh_grid(a, b, N, γ) -> Any
    Create a nonuniform grid of N + 1 points from a to b, as proposed by Trias et al. [10].
    source
    Setup ​
    KernelAbstractions.CPU Type
    julia
    struct CPU <: KernelAbstractions.Backend
    CPU(; static=false)
    Instantiate a CPU (multi-threaded) backend.
    Options:
    • static: Uses a static thread assignment, this can be beneficial for NUMA aware code. Defaults to false.
    Fields
    • static
    source
    IncompressibleNavierStokes.Setup Method
    julia
    Setup(
+3
    Fields
    source
    IncompressibleNavierStokes.Grid Method
    julia
    Grid(; x, boundary_conditions, backend)
    Create useful quantities for Cartesian box mesh ``x[1] \times \dots \times x[d]with boundary conditionsboundary_conditions. Return a named tuple ([α]` denotes a tuple index) with the following fields:
    • N[α]: Number of finite volumes in direction β, including ghost volumes
    • Nu[α][β]: Number of u[α] velocity DOFs in direction β
    • Np[α]: Number of pressure DOFs in direction α
    • Iu[α]: Cartesian index range of u[α] velocity DOFs
    • Ip: Cartesian index range of pressure DOFs
    • xlims[α]: Tuple containing the limits of the physical domain (not grid) in the direction α
    • x[α]: α-coordinates of all volume boundaries, including the left point of the first ghost volume
    • xu[α][β]: β-coordinates of u[α] velocity points
    • xp[α]: α-coordinates of pressure points
    • Δ[α]: All volume widths in direction α
    • Δu[α]: Distance between pressure points in direction α
    • A[α][β]: Interpolation weights from α-face centers xI to xI±hβ
    Note that the memory footprint of the redundant 1D-arrays above is negligible compared to the memory footprint of the 2D/3D-fields used in the code.
    source
    IncompressibleNavierStokes.cosine_grid Method
    julia
    cosine_grid(a, b, N) -> Any
    Create a nonuniform grid of N + 1 points from a to b using a cosine profile, i.e.
    xi=a+12(1−cos⁡(πin))(b−a),i=0,…,N
    See also stretched_grid.
    source
    IncompressibleNavierStokes.max_size Method
    julia
    max_size(grid) -> Any
    Get size of the largest grid element.
    source
    IncompressibleNavierStokes.stretched_grid Function
    julia
    stretched_grid(a, b, N) -> Any
+stretched_grid(a, b, N, s) -> Any
    Create a nonuniform grid of N + 1 points from a to b with a stretch factor of s. If s = 1, return a uniform spacing from a to b. Otherwise, return a vector x∈RN+1 such that xn=a+∑i=1nsi−1h for n=0,…,N. Setting xN=b then gives h=(b−a)1−s1−sN, resulting in
    xn=a+(b−a)1−sn1−sN,n=0,…,N.
    Note that stretched_grid(a, b, N, s)[n] corresponds to xn−1.
    See also cosine_grid.
    source
    IncompressibleNavierStokes.tanh_grid Function
    julia
    tanh_grid(a, b, N) -> Any
+tanh_grid(a, b, N, γ) -> Any
    Create a nonuniform grid of N + 1 points from a to b, as proposed by Trias et al. [10].
    source
    Setup ​
    KernelAbstractions.CPU Type
    julia
    struct CPU <: KernelAbstractions.Backend
    CPU(; static=false)
    Instantiate a CPU (multi-threaded) backend.
    Options:
    • static: Uses a static thread assignment, this can be beneficial for NUMA aware code. Defaults to false.
    Fields
    • static
    source
    IncompressibleNavierStokes.Setup Method
    julia
    Setup(
 ;
     x,
     boundary_conditions,
@@ -41,7 +41,7 @@
     workgroupsize,
     temperature,
     Re
-)
    Create problem setup (stored in a named tuple).
    source
    IncompressibleNavierStokes.temperature_equation Method
    julia
    temperature_equation(
+)
    Create problem setup (stored in a named tuple).
    source
    IncompressibleNavierStokes.temperature_equation Method
    julia
    temperature_equation(
 ;
     Pr,
     Ra,
@@ -50,11 +50,11 @@
     boundary_conditions,
     gdir,
     nondim_type
-)
    Create temperature equation setup (stored in a named tuple).
    The equation is parameterized by three dimensionless numbers (Prandtl number, Rayleigh number, and Gebhart number), and requires separate boundary conditions for the temperature field. The gdir keyword specifies the direction gravity, while non_dim_type specifies the type of non-dimensionalization.
    source
    Field initializers ​
    IncompressibleNavierStokes.random_field Function
    julia
    random_field(setup; ...) -> Any
-random_field(setup, t; A, kp, psolver, rng) -> Any
    Create random field, as in [11].
    • A: Eddy amplitude scaling
    • kp: Peak energy wavenumber
    source
    IncompressibleNavierStokes.scalarfield Method
    julia
    scalarfield(setup) -> Any
    Create empty scalar field.
    source
    IncompressibleNavierStokes.temperaturefield Function
    julia
    temperaturefield(setup, tempfunc) -> Any
-temperaturefield(setup, tempfunc, t) -> Any
    Create temperature field from function with boundary conditions at time t.
    source
    IncompressibleNavierStokes.vectorfield Method
    julia
    vectorfield(setup) -> Any
    Create empty vector field.
    source
    IncompressibleNavierStokes.velocityfield Function
    julia
    velocityfield(setup, ufunc; ...) -> Any
-velocityfield(setup, ufunc, t; psolver, doproject) -> Any
    Create divergence free velocity field u with boundary conditions at time t. The initial conditions of u[α] are specified by the function ufunc(α, x...).
    source
    
-    
+)

    Create temperature equation setup (stored in a named tuple).

    The equation is parameterized by three dimensionless numbers (Prandtl number, Rayleigh number, and Gebhart number), and requires separate boundary conditions for the temperature field. The gdir keyword specifies the direction gravity, while non_dim_type specifies the type of non-dimensionalization.

    source

    Field initializers ​

    IncompressibleNavierStokes.random_field Function
    julia
    random_field(setup; ...) -> Any
+random_field(setup, t; A, kp, psolver, rng) -> Any

    Create random field, as in [11].

    • A: Eddy amplitude scaling

    • kp: Peak energy wavenumber

    source

    IncompressibleNavierStokes.scalarfield Method
    julia
    scalarfield(setup) -> Any

    Create empty scalar field.

    source

    IncompressibleNavierStokes.temperaturefield Function
    julia
    temperaturefield(setup, tempfunc) -> Any
+temperaturefield(setup, tempfunc, t) -> Any

    Create temperature field from function with boundary conditions at time t.

    source

    IncompressibleNavierStokes.vectorfield Method
    julia
    vectorfield(setup) -> Any

    Create empty vector field.

    source

    IncompressibleNavierStokes.velocityfield Function
    julia
    velocityfield(setup, ufunc; ...) -> Any
+velocityfield(setup, ufunc, t; psolver, doproject) -> Any

    Create divergence free velocity field u with boundary conditions at time t. The initial conditions of u[α] are specified by the function ufunc(α, x...).

    source

    + \ No newline at end of file diff --git a/dev/manual/solver.html b/dev/manual/solver.html index 4c9b7558d..cad44a4b7 100644 --- a/dev/manual/solver.html +++ b/dev/manual/solver.html @@ -9,11 +9,11 @@ - + - + - + @@ -23,9 +23,9 @@ -
    IncompressibleNavierStokes.jl

    Appearance

    Solvers ​

    Solvers ​

    IncompressibleNavierStokes.get_cfl_timestep! Method
    julia
    get_cfl_timestep!(buf, u, setup) -> Any

    Get proposed maximum time step for convection and diffusion terms.

    source

    IncompressibleNavierStokes.get_state Method
    julia
    get_state(
+    
    IncompressibleNavierStokes.jl
    Appearance
    Solvers ​
    Solvers ​
    IncompressibleNavierStokes.get_cfl_timestep! Method
    julia
    get_cfl_timestep!(buf, u, setup) -> Any
    Get proposed maximum time step for convection and diffusion terms.
    source
    IncompressibleNavierStokes.get_state Method
    julia
    get_state(
     stepper
-) -> NamedTuple{(:u, :temp, :t, :n), <:NTuple{4, Any}}
    Get state (; u, temp, t, n) from stepper.
    source
    IncompressibleNavierStokes.solve_unsteady Method
    julia
    solve_unsteady(
+) -> NamedTuple{(:u, :temp, :t, :n), <:NTuple{4, Any}}
    Get state (; u, temp, t, n) from stepper.
    source
    IncompressibleNavierStokes.solve_unsteady Method
    julia
    solve_unsteady(
 ;
     setup,
     tlims,
@@ -41,7 +41,7 @@
     processors,
     θ,
     cache
-)
    Solve unsteady problem using method.
    If Δt is a real number, it is rounded such that (t_end - t_start) / Δt is an integer. If Δt = nothing, the time step is chosen every n_adapt_Δt iteration with CFL-number cfl. If Δt_min is given, the adaptive time step never goes below it.
    The processors are called after every time step.
    Note that the state observable passed to the processor.initialize function contains vector living on the device, and you may have to move them back to the host using Array(u) in the processor.
    Return (; u, t), outputs, where outputs is a named tuple with the outputs of processors with the same field names.
    source
    Processors ​
    Processors can be used to process the solution in solve_unsteady after every time step.
    IncompressibleNavierStokes.animator Function
    Animate a plot of the solution every update iteration. The animation is saved to path, which should have one of the following extensions:
    • ".mkv"
    • ".mp4"
    • ".webm"
    • ".gif"
    The plot is determined by a plotter processor. Additional kwargs are passed to plot.
    source
    IncompressibleNavierStokes.energy_history_plot Function
    Create energy history plot.
    source
    IncompressibleNavierStokes.energy_spectrum_plot Function
    Create energy spectrum plot. The energy at a scalar wavenumber level κ∈N is defined by
    e^(κ)=∫κ≤∥k∥2<κ+1|e^(k)|dk,
    as in San and Staples [12].
    Keyword arguments:
    • sloperange = [0.6, 0.9]: Percentage (between 0 and 1) of x-axis where the slope is plotted.
    • slopeoffset = 1.3: How far above the energy spectrum the inertial slope is plotted.
    • kwargs...: They are passed to observespectrum.
    source
    IncompressibleNavierStokes.fieldplot Function
    Plot state field in pressure points. If state is Observable, then the plot is interactive.
    Available fieldnames are:
    • :velocity,
    • :vorticity,
    • :streamfunction,
    • :pressure.
    Available plot types for 2D are:
    • heatmap (default),
    • image,
    • contour,
    • contourf.
    Available plot types for 3D are:
    • contour (default).
    The alpha value gets passed to contour in 3D.
    source
    IncompressibleNavierStokes.fieldsaver Method
    julia
    fieldsaver(; setup, nupdate)
    Create processor that stores the solution and time every nupdate time step.
    source
    IncompressibleNavierStokes.observefield Method
    julia
    observefield(state; setup, fieldname, logtol, psolver)
    Observe field fieldname at pressure points.
    source
    IncompressibleNavierStokes.observespectrum Method
    julia
    observespectrum(state; setup, npoint, a)
    Observe energy spectrum of state.
    source
    IncompressibleNavierStokes.processor Function
    julia
    processor(
+)
    Solve unsteady problem using method.
    If Δt is a real number, it is rounded such that (t_end - t_start) / Δt is an integer. If Δt = nothing, the time step is chosen every n_adapt_Δt iteration with CFL-number cfl. If Δt_min is given, the adaptive time step never goes below it.
    The processors are called after every time step.
    Note that the state observable passed to the processor.initialize function contains vector living on the device, and you may have to move them back to the host using Array(u) in the processor.
    Return (; u, t), outputs, where outputs is a named tuple with the outputs of processors with the same field names.
    source
    Processors ​
    Processors can be used to process the solution in solve_unsteady after every time step.
    IncompressibleNavierStokes.animator Function
    Animate a plot of the solution every update iteration. The animation is saved to path, which should have one of the following extensions:
    • ".mkv"
    • ".mp4"
    • ".webm"
    • ".gif"
    The plot is determined by a plotter processor. Additional kwargs are passed to plot.
    source
    IncompressibleNavierStokes.energy_history_plot Function
    Create energy history plot.
    source
    IncompressibleNavierStokes.energy_spectrum_plot Function
    Create energy spectrum plot. The energy at a scalar wavenumber level κ∈N is defined by
    e^(κ)=∫κ≤∥k∥2<κ+1|e^(k)|dk,
    as in San and Staples [12].
    Keyword arguments:
    • sloperange = [0.6, 0.9]: Percentage (between 0 and 1) of x-axis where the slope is plotted.
    • slopeoffset = 1.3: How far above the energy spectrum the inertial slope is plotted.
    • kwargs...: They are passed to observespectrum.
    source
    IncompressibleNavierStokes.fieldplot Function
    Plot state field in pressure points. If state is Observable, then the plot is interactive.
    Available fieldnames are:
    • :velocity,
    • :vorticity,
    • :streamfunction,
    • :pressure.
    Available plot types for 2D are:
    • heatmap (default),
    • image,
    • contour,
    • contourf.
    Available plot types for 3D are:
    • contour (default).
    The alpha value gets passed to contour in 3D.
    source
    IncompressibleNavierStokes.fieldsaver Method
    julia
    fieldsaver(; setup, nupdate)
    Create processor that stores the solution and time every nupdate time step.
    source
    IncompressibleNavierStokes.observefield Method
    julia
    observefield(state; setup, fieldname, logtol, psolver)
    Observe field fieldname at pressure points.
    source
    IncompressibleNavierStokes.observespectrum Method
    julia
    observespectrum(state; setup, npoint, a)
    Observe energy spectrum of state.
    source
    IncompressibleNavierStokes.processor Function
    julia
    processor(
     initialize
 ) -> NamedTuple{(:initialize, :finalize), <:Tuple{Any, IncompressibleNavierStokes.var"#295#296"}}
 processor(
@@ -65,7 +65,7 @@
 The summand is 3, the time is 1.2
 The summand is 4, the time is 1.6
 The summand is 5, the time is 2.0
-The final sum (at time t=2.0) is 15
    source
    IncompressibleNavierStokes.realtimeplotter Function
    Processor for plotting the solution in real time.
    Keyword arguments:
    • plot: Plot function.
    • nupdate: Show solution every nupdate time step.
    • displayfig: Display the figure at the start.
    • screen: If nothing, use default display. If GLMakie.screen() multiple plots can be displayed in separate windows like in MATLAB (see also GLMakie.closeall()).
    • displayupdates: Display the figure at every update (if using CairoMakie).
    • sleeptime: The sleeptime is slept at every update, to give Makie time to update the plot. Set this to nothing to skip sleeping.
    Additional kwargs are passed to the plot function.
    source
    IncompressibleNavierStokes.save_vtk Method
    julia
    save_vtk(state; setup, filename, kwargs...)
    Save fields to vtk file.
    The kwargs are passed to snapshotsaver.
    source
    IncompressibleNavierStokes.snapshotsaver Method
    julia
    snapshotsaver(state; setup, fieldnames, psolver)
    In the case of a 2D setup, the velocity field is saved as a 3D vector with a z-component of zero, as this seems to be preferred by ParaView.
    source
    IncompressibleNavierStokes.timelogger Method
    julia
    timelogger(
+The final sum (at time t=2.0) is 15
    source
    IncompressibleNavierStokes.realtimeplotter Function
    Processor for plotting the solution in real time.
    Keyword arguments:
    • plot: Plot function.
    • nupdate: Show solution every nupdate time step.
    • displayfig: Display the figure at the start.
    • screen: If nothing, use default display. If GLMakie.screen() multiple plots can be displayed in separate windows like in MATLAB (see also GLMakie.closeall()).
    • displayupdates: Display the figure at every update (if using CairoMakie).
    • sleeptime: The sleeptime is slept at every update, to give Makie time to update the plot. Set this to nothing to skip sleeping.
    Additional kwargs are passed to the plot function.
    source
    IncompressibleNavierStokes.save_vtk Method
    julia
    save_vtk(state; setup, filename, kwargs...)
    Save fields to vtk file.
    The kwargs are passed to snapshotsaver.
    source
    IncompressibleNavierStokes.snapshotsaver Method
    julia
    snapshotsaver(state; setup, fieldnames, psolver)
    In the case of a 2D setup, the velocity field is saved as a 3D vector with a z-component of zero, as this seems to be preferred by ParaView.
    source
    IncompressibleNavierStokes.timelogger Method
    julia
    timelogger(
 ;
     showiter,
     showt,
@@ -73,8 +73,8 @@
     showmax,
     showspeed,
     nupdate
-) -> @NamedTuple{initialize::IncompressibleNavierStokes.var"#298#300"{Bool, Bool, Bool, Bool, Bool, Int64}, finalize::IncompressibleNavierStokes.var"#295#296"}
    Create processor that logs time step information.
    source
    IncompressibleNavierStokes.vtk_writer Method
    julia
    vtk_writer(; setup, nupdate, dir, filename, kwargs...)
    Create processor that writes the solution every nupdate time steps to a VTK file. The resulting Paraview data collection file is stored in "$dir/$filename.pvd". The kwargs are passed to snapshotsaver.
    source
    
-    
+) -> @NamedTuple{initialize::IncompressibleNavierStokes.var"#298#300"{Bool, Bool, Bool, Bool, Bool, Int64}, finalize::IncompressibleNavierStokes.var"#295#296"}

    Create processor that logs time step information.

    source

    IncompressibleNavierStokes.vtk_writer Method
    julia
    vtk_writer(; setup, nupdate, dir, filename, kwargs...)

    Create processor that writes the solution every nupdate time steps to a VTK file. The resulting Paraview data collection file is stored in "$dir/$filename.pvd". The kwargs are passed to snapshotsaver.

    source

    + \ No newline at end of file diff --git a/dev/manual/spatial.html b/dev/manual/spatial.html index 5c38e2e88..3969032e9 100644 --- a/dev/manual/spatial.html +++ b/dev/manual/spatial.html @@ -9,9 +9,9 @@ - + - + @@ -24,7 +24,7 @@
    IncompressibleNavierStokes.jl

    Appearance

    Spatial discretization ​

    the d spatial dimensions are indexed by α∈{1,…,d}. The α-th unit vector is denoted eα=(eαβ)β=1d, where the Kronecker symbol eαβ is 1 if α=β and 0 otherwise. We note hα=eα/2. The Cartesian index I=(I1,…,Id) is used to avoid repeating terms and equations d times, where Iα is a scalar index (typically one of i, j, and k in common notation). This notation is dimension-agnostic, since we can write uI instead of uij in 2D or uijk in 3D. In our Julia implementation of the solver we use the same Cartesian notation (u[I] instead of u[i, j] or u[i, j, k]).

    For the discretization scheme, we use a staggered Cartesian grid as proposed by Harlow and Welch [13]. Consider a rectangular domain Ω=∏α=1d[aα,bα], where aα<bα are the domain boundaries and ∏ is a Cartesian product. Let Ω=⋃I∈IΩI be a partitioning of Ω, where I=∏α=1d{12,2−12,…,Nα−12} are volume center indices, N=(N1,…,Nd)∈Nd are the number of volumes in each dimension, ΩI=∏α=1dΔIαα is a finite volume, ΓIα=ΩI−hα∩ΩI+hα=∏β≠αΔIββ is a volume face, Δiα=[xi−12α,xi+12α] is a volume edge, x0α,…,xNαα are volume boundary coordinates, and xiα=12(xi−12α+xi+12α) for i∈{1/2,…,Nα−1/2} are volume center coordinates. We also define the operator δα which maps a discrete scalar field φ=(φI)I to

    (δαφ)I=φI+hα−φI−hα|ΔIαα|.

    It can be interpreted as a discrete equivalent of the continuous operator ∂∂xα. All the above definitions are extended to be valid in volume centers I∈I, volume faces I∈I+hα, or volume corners I∈I+∑α=1dhα. The discretization is illustrated below:

    Finite volume discretization of the Navier-Stokes equations ​

    We now define the unknown degrees of freedom. The average pressure in ΩI, I∈I is approximated by the quantity pI(t). The average α-velocity on the face ΓIα, I∈I+hα is approximated by the quantity uIα(t). Note how the pressure p and the d velocity fields uα are each defined in their own canonical positions xI and xI+hα for I∈I. In the following, we derive equations for these unknowns.

    Using the pressure control volume O=ΩI with I∈I in the mass integral constraint and approximating the face integrals with the mid-point quadrature rule ∫ΓIudΓ≈|ΓI|uI results in the discrete divergence-free constraint

    ∑α=1d(δαuα)I=0.

    Note how dividing by the volume size results in a discrete equation resembling the continuous one (since |ΩI|=|ΓIα||ΔIαα|).

    Similarly, choosing an α-velocity control volume O=ΩI with I∈I+hα in the integral momentum equation, approximating the volume- and face integrals using the mid-point quadrature rule, and replacing remaining spatial derivatives in the diffusive term with a finite difference approximation gives the discrete momentum equations

    ddtuIα=−∑β=1d(δβ(uαuβ))I+ν∑β=1d(δβδβuα)I+fα(xI)−(δαp)I.

    where we made the assumption that f is constant in time for simplicity. The outer discrete derivative in (δβδβuα)I is required at the position I, which means that the inner derivative is evaluated as (δβuα)I+hβ and (δβuα)I−hβ, thus requiring uI−2hβα, uIα, and uI+2hβα, which are all in their canonical positions. The two velocity components in the convective term uαuβ are required at the positions I−hβ and I+hβ, which are outside the canonical positions. Their value at the required position is obtained using averaging with weights 1/2 for the α-component and with linear interpolation for the β-component. This preserves the skew-symmetry of the convection operator, such that energy is conserved (in the convective term) [14].

    Boundary conditions ​

    Storage convention

    We use the column-major convention (Julia, MATLAB, Fortran), and not the row-major convention (Python, C). Thus the x1-index i will vary for one whole cycle in the vectors before the x2-index j, x3 index k, and component-index α are incremented, e.g. uh=(u(1,1,1)1,u(2,1,1)1,…u(Nu3(1),Nu3(2),Nu3(3))3) in 3D.

    Fourth order accurate discretization ​

    The above discretization is second order accurate. A fourth order accurate discretization can be obtained by judiciously combining the second order discretization with itself on a grid with three times larger cells in each dimension [14] [15]. The coarse discretization is identical, but the mass equation is derived for the three times coarser control volume

    ΩI3=⋃α=1dΩI−eα∪ΩI∪ΩI+eα,

    while the momentum equation is derived for its shifted variant ΩI+hα3. The resulting fourth order accurate equations are given by

    ∑α=1d(δαuα)I−|ΩI3|32+d|ΩI|∑α=1d(δα3uα)I=0

    and

    ddtuIα=−∑β=1d(δβ(uαuβ))I+ν∑β=1d(δβδβuα)I+fα(xI)−(δαp)I+fourth order,

    where

    (δα3φ)I=φI+3hα−φI−3hαΔIα−1α+ΔIαα+ΔIα+1α.

    Matrix representation ​

    We can write the mass and momentum equations in matrix form. We will use the same matrix notation for the second- and fourth order accurate discretizations. The discrete mass equation becomes

    Muh+yM=0,

    where M is the discrete divergence operator and yM contains the boundary value contributions of the velocity to the divergence field.

    The discrete momentum equations become

    duhdt=−C(uh)+ν(Duh+yD)+fh−(Gph+yG)=F(uh)−(Gph+yG),

    where C is she convection operator (including boundary contributions), D is the diffusion operator, yD is boundary contribution to the diffusion term, G=Wu−1MTW is the pressure gradient operator, yG contains the boundary contribution of the pressure to the pressure gradient (only non-zero for pressure boundary conditions), Wu is a diagonal matrix containing the velocity volume sizes |ΩI+δ(α)/2|, and W is a diagonal matrix containing the reference volume sizes |ΩI|. The term F refers to all the forces except for the pressure gradient.

    Volume normalization

    All the operators have been divided by the velocity volume sizes. As a result, the operators have the same units as their continuous counterparts.

    Discrete pressure Poisson equation ​

    Instead of directly discretizing the continuous pressure Poisson equation, we will rededuce it in the discrete setting, thus aiming to preserve the discrete divergence freeness instead of the continuous one. Applying the discrete divergence operator M to the discrete momentum equations yields the discrete pressure Poisson equation

    Lph=WM(F(uh)−yG)+WdyMdt,

    where L=WMG=WMWu−1MTW is a discrete Laplace operator. It is positive symmetric.

    Unsteady Dirichlet boundary conditions

    If the equations are prescribed with unsteady Dirichlet boundary conditions, for example an inflow that varies with time, the term dyMdt will be non-zero. If this term is not known exactly, for example if the next value of the inflow is unknown at the time of the current value, it must be computed using past values of of the velocity inflow only, for example dyMdt≈(yM(t)−yM(t−Δt))/Δt for some Δt.

    Uniqueness of pressure field

    Unless pressure boundary conditions are present, the pressure is only determined up to a constant, as L will have an eigenvalue of zero. Since only the gradient of the pressure appears in the equations, we can set the unknown constant to zero without affecting the velocity field.

    Pressure projection

    The pressure field ph can be seen as a Lagrange multiplier enforcing the constraint of discrete divergence freeness. It is also possible to write the momentum equations without the pressure by explicitly solving the discrete Poisson equation:

    ph=L−1WM(F(uh)−yG)+L−1WdyMdt.

    The momentum equations then become

    duhdt=(I−GL−1WM)(F(uh)−yG)−GL−1WdyMdt.

    The matrix (I−GL−1WM) is a projector onto the space of discretely divergence free velocities. However, using this formulation would require an efficient way to perform the projection without assembling the operator matrix L−1, which would be very costly.

    Discrete output quantities ​

    Kinetic energy ​

    The local kinetic energy is defined by k=12∥u∥22=12∑α=1duαuα. On the staggered grid however, the different velocity components are not located at the same point. We will therefore interpolate the velocity to the pressure point before summing the squares.

    Vorticity ​

    In 2D, the vorticity is a scalar. We define it as

    ω=−δ2u1+δ1u2.

    The 3D vorticity is a vector field (ω1,ω2,ω3). Noting α+=mod3(α+1) and α−=mod3(α−1), the vorticity is defined as through

    ωα=−δα−uα++δα+uα−.

    Stream function ​

    In 2D, the stream function is defined at the corners with the vorticity. Integrating the stream function Poisson equation over the vorticity volume yields

    −∫ΩI+h1+h2ωdΩ=∫ΩI+h1+h2∇2ψdΩ=∫ΓI+e1+h21∂ψ∂x1dΓ−∫ΓI+h21∂ψ∂x1dΓ+∫ΓI+h1+e22∂ψ∂x2dΓ−∫ΓI+h12∂ψ∂x2dΓ.

    Replacing the integrals with the mid-point quadrature rule and the spatial derivatives with central finite differences yields the discrete Poisson equation for the stream function at the vorticity point:

    |ΓI+h1+h21|(ψI+3/h1+h2−ψI+h1+h2xI1+3/21−xI1+1/21−ψI+h1+h2−ψI−h1+h2xI1+1/21−xI1−1/21)+|ΓI+h1+h22|(ψI+h1+3h2−ψI+h1+h2xI1+3/22−xI1+1/22−ψI+h1+h2−ψI+h1−h2xI2+1/22−xI2−1/22)=|ΩI+h1+h2|ωI+h1+h2
    - + \ No newline at end of file diff --git a/dev/manual/temperature.html b/dev/manual/temperature.html index 962ce6175..0b7810706 100644 --- a/dev/manual/temperature.html +++ b/dev/manual/temperature.html @@ -9,11 +9,11 @@ - + - + - + @@ -35,8 +35,8 @@ boundary_conditions, gdir, nondim_type -)

    Create temperature equation setup (stored in a named tuple).

    The equation is parameterized by three dimensionless numbers (Prandtl number, Rayleigh number, and Gebhart number), and requires separate boundary conditions for the temperature field. The gdir keyword specifies the direction gravity, while non_dim_type specifies the type of non-dimensionalization.

    source

    - +)

    Create temperature equation setup (stored in a named tuple).

    The equation is parameterized by three dimensionless numbers (Prandtl number, Rayleigh number, and Gebhart number), and requires separate boundary conditions for the temperature field. The gdir keyword specifies the direction gravity, while non_dim_type specifies the type of non-dimensionalization.

    source

    + \ No newline at end of file diff --git a/dev/manual/time.html b/dev/manual/time.html index 25ce43936..fccc586f2 100644 --- a/dev/manual/time.html +++ b/dev/manual/time.html @@ -9,11 +9,11 @@ - + - + - + @@ -23,181 +23,181 @@ -
    IncompressibleNavierStokes.jl

    Appearance

    Time discretization ​

    The spatially discretized Navier-Stokes equations form a differential-algebraic system, with an ODE for the velocity

    dudt=F(u,t)−(Gp+yG)

    subject to the algebraic constraint formed by the mass equation

    Mu+yM=0.

    In the end of the previous section, we differentiated the mass equation in time to obtain a discrete pressure Poisson equation. This equation includes the term dyMdt, which is non-zero if an unsteady flow of mass is added to the domain (Dirichlet boundary conditions). This term ensures that the time-continuous discrete velocity field u(t) stays divergence free (conserves mass). However, if we directly discretize this system in time, the mass preservation may actually not be respected. For this, we will change the definition of the pressure such that the time-discretized velocity field is divergence free at each time step and each time sub-step (to be defined in the following).

    Consider the interval [0,T] for some simulation time T. We will divide it into N sub-intervals [tn,tn+1] for n=0,…,N−1, with t0=0, tN=T, and increment Δtn=tn+1−tn. We define un≈u(tn) as an approximation to the exact discrete velocity field u(tn), with u0=u(0) starting from the exact initial conditions. We say that the time integration scheme (definition of un) is accurate to the order r if un=u(tn)+O(Δtr) for all n.

    IncompressibleNavierStokes provides a collection of explicit and implicit Runge-Kutta methods, in addition to Adams-Bashforth Crank-Nicolson and one-leg beta method time steppers.

    The code is currently not adapted to time steppers from DifferentialEquations.jl, but they may be integrated in the future.

    IncompressibleNavierStokes.AbstractODEMethod Type
    julia
    abstract type AbstractODEMethod{T}

    Abstract ODE method.

    Fields

    source

    IncompressibleNavierStokes.ode_method_cache Function
    julia
    ode_method_cache(method, setup, u, temp)

    Get time stepper cache for the given ODE method.

    source

    IncompressibleNavierStokes.runge_kutta_method Function
    julia
    runge_kutta_method(
+    
    IncompressibleNavierStokes.jl
    Appearance
    Time discretization ​
    The spatially discretized Navier-Stokes equations form a differential-algebraic system, with an ODE for the velocity
    dudt=F(u,t)−(Gp+yG)
    subject to the algebraic constraint formed by the mass equation
    Mu+yM=0.
    In the end of the previous section, we differentiated the mass equation in time to obtain a discrete pressure Poisson equation. This equation includes the term dyMdt, which is non-zero if an unsteady flow of mass is added to the domain (Dirichlet boundary conditions). This term ensures that the time-continuous discrete velocity field u(t) stays divergence free (conserves mass). However, if we directly discretize this system in time, the mass preservation may actually not be respected. For this, we will change the definition of the pressure such that the time-discretized velocity field is divergence free at each time step and each time sub-step (to be defined in the following).
    Consider the interval [0,T] for some simulation time T. We will divide it into N sub-intervals [tn,tn+1] for n=0,…,N−1, with t0=0, tN=T, and increment Δtn=tn+1−tn. We define un≈u(tn) as an approximation to the exact discrete velocity field u(tn), with u0=u(0) starting from the exact initial conditions. We say that the time integration scheme (definition of un) is accurate to the order r if un=u(tn)+O(Δtr) for all n.
    IncompressibleNavierStokes provides a collection of explicit and implicit Runge-Kutta methods, in addition to Adams-Bashforth Crank-Nicolson and one-leg beta method time steppers.
    The code is currently not adapted to time steppers from DifferentialEquations.jl, but they may be integrated in the future.
    IncompressibleNavierStokes.AbstractODEMethod Type
    julia
    abstract type AbstractODEMethod{T}
    Abstract ODE method.
    Fields
    source
    IncompressibleNavierStokes.ode_method_cache Function
    julia
    ode_method_cache(method, setup, u, temp)
    Get time stepper cache for the given ODE method.
    source
    IncompressibleNavierStokes.runge_kutta_method Function
    julia
    runge_kutta_method(
     A,
     b,
     c,
     r;
     T,
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod, IncompressibleNavierStokes.ImplicitRungeKuttaMethod}
    Get Runge Kutta method. The function checks whether the method is explicit.
    p_add_solve: whether to add a pressure solve step to the method.
    For implicit RK methods: newton_type, maxiter, abstol, reltol.
    source
    IncompressibleNavierStokes.create_stepper Function
    julia
    create_stepper(method; setup, psolver, u, temp, t, n)
    Create time stepper.
    source
    IncompressibleNavierStokes.timestep Function
    julia
    timestep(method, stepper, Δt; θ = nothing)
    Perform one time step.
    Non-mutating/allocating/out-of-place version.
    See also timestep!.
    source
    IncompressibleNavierStokes.timestep! Function
    julia
    timestep!(method, stepper, Δt; θ = nothing, cache)
    Perform one time step>
    Mutating/non-allocating/in-place version.
    See also timestep.
    source
    Adams-Bashforth Crank-Nicolson method ​
    IncompressibleNavierStokes.AdamsBashforthCrankNicolsonMethod Type
    julia
    struct AdamsBashforthCrankNicolsonMethod{T, M} <: IncompressibleNavierStokes.AbstractODEMethod{T}
    IMEX AB-CN: Adams-Bashforth for explicit convection (parameters α₁ and α₂) and Crank-Nicolson for implicit diffusion (implicitness θ). The method is second order for θ = 1/2.
    The LU decomposition of the LHS matrix is computed every time the time step changes.
    Note that, in contrast to explicit methods, the pressure from previous time steps has an influence on the accuracy of the velocity.
    We here require that the time step Δt is constant. This methods uses Adams-Bashforth for the convective terms and Crank-Nicolson stepping for the diffusion and body force terms. Given the velocity field u0=u(t0) at a time t0 and its previous value u−1=u(t0−Δt) at the previous time t−1=t0−Δt, the predicted velocity field u at the time t=t0+Δt is defined by first computing a tentative velocity:
    v−u0Δt=−(α0C(u0,t0)+α−1C(u−1,t−1))+θ(Du0+yD(t0))+(1−θ)(Dv+yD(t))+θf(t0)+(1−θ)f(t)−(Gp0+yG(t0)),
    where θ∈[0,1] is the Crank-Nicolson parameter (θ=12 for second order convergence), (α0,α−1)=(32,−12) are the Adams-Bashforth coefficients, and v is a tentative velocity yet to be made divergence free. We can group the terms containing v on the left hand side, to obtain
    (1ΔtI−(1−θ)D)v=(1ΔtI−θD)u0−(α0C(u0,t0)+α−1C(u−1,t−1))+θyD(t0)+(1−θ)yD(t)+θf(t0)+(1−θ)f(t)−(Gp0+yG(t0)).
    We can compute v by inverting the positive definite matrix (1ΔtI−θD) for the given right hand side using a suitable linear solver. Assuming Δt is constant, we can precompute a Cholesky factorization of this matrix before starting time stepping.
    We then compute the pressure difference Δp by solving
    LΔp=WMv+yM(t)Δt−WM(yG(t)−yG(t0)),
    after which a divergence free velocity u can be enforced:
    u=v−Δt(GΔp+yG(t)−yG(t0)).
    A first order accurate prediction of the corresponding pressure is p=p0+Δp. However, since this pressure is reused in the next time step, we perform an additional pressure solve to avoid accumulating first order errors. The resulting pressure p is then accurate to the same order as u.
    Fields
    • α₁
    • α₂
    • θ
    • p_add_solve
    • method_startup
    source
    One-leg beta method ​
    IncompressibleNavierStokes.OneLegMethod Type
    julia
    struct OneLegMethod{T, M} <: IncompressibleNavierStokes.AbstractODEMethod{T}
    Explicit one-leg β-method following symmetry-preserving discretization of turbulent flow. See Verstappen and Veldman [14] [16] for details.
    We here require that the time step Δt is constant. Given the velocity u0 and pressure p0 at the current time t0 and their previous values u−1 and p−1 at the time t−1=t0−Δt, we start by computing the "offstep" values v=(1+β)v0−βv−1 and Q=(1+β)p0−βp−1 for some β=12.
    A tentative velocity field v~ is then computed as follows:
    v~=1β+12(2βu0−(β−12)u−1+ΔtF(v,t)−Δt(GQ+yG(t))).
    A pressure correction Δp is obtained by solving the Poisson equation
    LΔp=β+12ΔtW(Mv~+yM(t)).
    Finally, the divergence free velocity field is given by
    u=v~−Δtβ+12GΔp,
    while the second order accurate pressure is given by
    p=2p0−p−1+43Δp.
    Fields
    • β
    • p_add_solve
    • method_startup
    source
    Runge-Kutta methods ​
    IncompressibleNavierStokes.AbstractRungeKuttaMethod Type
    julia
    abstract type AbstractRungeKuttaMethod{T} <: IncompressibleNavierStokes.AbstractODEMethod{T}
    Abstract Runge Kutta method.
    Fields
    source
    IncompressibleNavierStokes.ExplicitRungeKuttaMethod Type
    julia
    struct ExplicitRungeKuttaMethod{T} <: IncompressibleNavierStokes.AbstractRungeKuttaMethod{T}
    Explicit Runge Kutta method. See Sanderse [17].
    Consider the velocity field u0 at a certain time t0. We will now perform one time step to t=t0+Δt. For explicit Runge-Kutta methods, this time step is divided into s sub-steps ti=t0+Δti with increment Δti=ciΔt. The final substep performs the full time step Δts=Δt such that ts=t.
    For i=1,…,s, the intermediate velocity ui and pressure pi are computed as follows:
    ki=F(ui−1,ti−1)−yG(ti−1)vi=u0+Δt∑j=1iaijkjLpi=WM1ci∑j=1iaijkj+WyM(ti)−yM(t0)Δti=W(Mvi+yM(ti))−(Mu0+yM(t0))Δtin=WMvi+yM(ti)Δtinui=vi−ΔtiGpi,
    where (aij)ij are the Butcher tableau coefficients of the RK-method, with the convention ci=∑j=1iaij.
    Finally, we return us. If u0=u(t0), we get the accuracy us=u(t)+O(Δtr+1), where r is the order of the RK-method. If we perform n RK time steps instead of one, starting at exact initial conditions u0=u(0), then un=u(tn)+O(Δtr) for all n∈{1,…,N}. Note that for a given u, the corresponding pressure p can be calculated to the same accuracy as u by doing an additional pressure projection after each outer time step Δt (if we know dyMdt(t)), or to first order accuracy by simply returning ps.
    Note that each of the sub-step velocities ui is divergence free, after projecting the tentative velocities vi. This is ensured due to the judiciously chosen replacement of dyMdt(ti) with (yM(ti)−yM(t0))/Δti. The space-discrete divergence-freeness is thus perfectly preserved, even though the time discretization introduces other errors.
    Fields
    • A
    • b
    • c
    • r
    • p_add_solve
    source
    IncompressibleNavierStokes.ImplicitRungeKuttaMethod Type
    julia
    struct ImplicitRungeKuttaMethod{T} <: IncompressibleNavierStokes.AbstractRungeKuttaMethod{T}
    Implicit Runge Kutta method. See Sanderse [18].
    The implicit linear system is solved at each time step using Newton's method. The newton_type may be one of the following:
    • :no: Replace iteration matrix with I/Δt (no Jacobian)
    • :approximate: Build Jacobian once before iterations only
    • :full: Build Jacobian at each iteration
    Fields
    • A
    • b
    • c
    • r
    • newton_type
    • maxiter
    • abstol
    • reltol
    • p_add_solve
    source
    IncompressibleNavierStokes.RKMethods Module
    Set up Butcher arrays A, b, and c, as well as and SSP coefficient r. For families of methods, optional input s is the number of stages.
    Original (MATLAB) by David Ketcheson, extended by Benjamin Sanderse.
    Exports
    source
    IncompressibleNavierStokes.LMWray3 Type
    julia
    struct LMWray3{T} <: IncompressibleNavierStokes.AbstractRungeKuttaMethod{T}
    Low memory Wray 3rd order scheme. Uses 3 vector fields and one scalar field.
    Fields
    source
    Explicit Methods ​
    IncompressibleNavierStokes.RKMethods.FE11 Function
    julia
    FE11(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod, IncompressibleNavierStokes.ImplicitRungeKuttaMethod}
    Get Runge Kutta method. The function checks whether the method is explicit.
    p_add_solve: whether to add a pressure solve step to the method.
    For implicit RK methods: newton_type, maxiter, abstol, reltol.
    source
    IncompressibleNavierStokes.create_stepper Function
    julia
    create_stepper(method; setup, psolver, u, temp, t, n)
    Create time stepper.
    source
    IncompressibleNavierStokes.timestep Function
    julia
    timestep(method, stepper, Δt; θ = nothing)
    Perform one time step.
    Non-mutating/allocating/out-of-place version.
    See also timestep!.
    source
    IncompressibleNavierStokes.timestep! Function
    julia
    timestep!(method, stepper, Δt; θ = nothing, cache)
    Perform one time step>
    Mutating/non-allocating/in-place version.
    See also timestep.
    source
    Adams-Bashforth Crank-Nicolson method ​
    IncompressibleNavierStokes.AdamsBashforthCrankNicolsonMethod Type
    julia
    struct AdamsBashforthCrankNicolsonMethod{T, M} <: IncompressibleNavierStokes.AbstractODEMethod{T}
    IMEX AB-CN: Adams-Bashforth for explicit convection (parameters α₁ and α₂) and Crank-Nicolson for implicit diffusion (implicitness θ). The method is second order for θ = 1/2.
    The LU decomposition of the LHS matrix is computed every time the time step changes.
    Note that, in contrast to explicit methods, the pressure from previous time steps has an influence on the accuracy of the velocity.
    We here require that the time step Δt is constant. This methods uses Adams-Bashforth for the convective terms and Crank-Nicolson stepping for the diffusion and body force terms. Given the velocity field u0=u(t0) at a time t0 and its previous value u−1=u(t0−Δt) at the previous time t−1=t0−Δt, the predicted velocity field u at the time t=t0+Δt is defined by first computing a tentative velocity:
    v−u0Δt=−(α0C(u0,t0)+α−1C(u−1,t−1))+θ(Du0+yD(t0))+(1−θ)(Dv+yD(t))+θf(t0)+(1−θ)f(t)−(Gp0+yG(t0)),
    where θ∈[0,1] is the Crank-Nicolson parameter (θ=12 for second order convergence), (α0,α−1)=(32,−12) are the Adams-Bashforth coefficients, and v is a tentative velocity yet to be made divergence free. We can group the terms containing v on the left hand side, to obtain
    (1ΔtI−(1−θ)D)v=(1ΔtI−θD)u0−(α0C(u0,t0)+α−1C(u−1,t−1))+θyD(t0)+(1−θ)yD(t)+θf(t0)+(1−θ)f(t)−(Gp0+yG(t0)).
    We can compute v by inverting the positive definite matrix (1ΔtI−θD) for the given right hand side using a suitable linear solver. Assuming Δt is constant, we can precompute a Cholesky factorization of this matrix before starting time stepping.
    We then compute the pressure difference Δp by solving
    LΔp=WMv+yM(t)Δt−WM(yG(t)−yG(t0)),
    after which a divergence free velocity u can be enforced:
    u=v−Δt(GΔp+yG(t)−yG(t0)).
    A first order accurate prediction of the corresponding pressure is p=p0+Δp. However, since this pressure is reused in the next time step, we perform an additional pressure solve to avoid accumulating first order errors. The resulting pressure p is then accurate to the same order as u.
    Fields
    • α₁
    • α₂
    • θ
    • p_add_solve
    • method_startup
    source
    One-leg beta method ​
    IncompressibleNavierStokes.OneLegMethod Type
    julia
    struct OneLegMethod{T, M} <: IncompressibleNavierStokes.AbstractODEMethod{T}
    Explicit one-leg β-method following symmetry-preserving discretization of turbulent flow. See Verstappen and Veldman [14] [16] for details.
    We here require that the time step Δt is constant. Given the velocity u0 and pressure p0 at the current time t0 and their previous values u−1 and p−1 at the time t−1=t0−Δt, we start by computing the "offstep" values v=(1+β)v0−βv−1 and Q=(1+β)p0−βp−1 for some β=12.
    A tentative velocity field v~ is then computed as follows:
    v~=1β+12(2βu0−(β−12)u−1+ΔtF(v,t)−Δt(GQ+yG(t))).
    A pressure correction Δp is obtained by solving the Poisson equation
    LΔp=β+12ΔtW(Mv~+yM(t)).
    Finally, the divergence free velocity field is given by
    u=v~−Δtβ+12GΔp,
    while the second order accurate pressure is given by
    p=2p0−p−1+43Δp.
    Fields
    • β
    • p_add_solve
    • method_startup
    source
    Runge-Kutta methods ​
    IncompressibleNavierStokes.AbstractRungeKuttaMethod Type
    julia
    abstract type AbstractRungeKuttaMethod{T} <: IncompressibleNavierStokes.AbstractODEMethod{T}
    Abstract Runge Kutta method.
    Fields
    source
    IncompressibleNavierStokes.ExplicitRungeKuttaMethod Type
    julia
    struct ExplicitRungeKuttaMethod{T} <: IncompressibleNavierStokes.AbstractRungeKuttaMethod{T}
    Explicit Runge Kutta method. See Sanderse [17].
    Consider the velocity field u0 at a certain time t0. We will now perform one time step to t=t0+Δt. For explicit Runge-Kutta methods, this time step is divided into s sub-steps ti=t0+Δti with increment Δti=ciΔt. The final substep performs the full time step Δts=Δt such that ts=t.
    For i=1,…,s, the intermediate velocity ui and pressure pi are computed as follows:
    ki=F(ui−1,ti−1)−yG(ti−1)vi=u0+Δt∑j=1iaijkjLpi=WM1ci∑j=1iaijkj+WyM(ti)−yM(t0)Δti=W(Mvi+yM(ti))−(Mu0+yM(t0))Δtin=WMvi+yM(ti)Δtinui=vi−ΔtiGpi,
    where (aij)ij are the Butcher tableau coefficients of the RK-method, with the convention ci=∑j=1iaij.
    Finally, we return us. If u0=u(t0), we get the accuracy us=u(t)+O(Δtr+1), where r is the order of the RK-method. If we perform n RK time steps instead of one, starting at exact initial conditions u0=u(0), then un=u(tn)+O(Δtr) for all n∈{1,…,N}. Note that for a given u, the corresponding pressure p can be calculated to the same accuracy as u by doing an additional pressure projection after each outer time step Δt (if we know dyMdt(t)), or to first order accuracy by simply returning ps.
    Note that each of the sub-step velocities ui is divergence free, after projecting the tentative velocities vi. This is ensured due to the judiciously chosen replacement of dyMdt(ti) with (yM(ti)−yM(t0))/Δti. The space-discrete divergence-freeness is thus perfectly preserved, even though the time discretization introduces other errors.
    Fields
    • A
    • b
    • c
    • r
    • p_add_solve
    source
    IncompressibleNavierStokes.ImplicitRungeKuttaMethod Type
    julia
    struct ImplicitRungeKuttaMethod{T} <: IncompressibleNavierStokes.AbstractRungeKuttaMethod{T}
    Implicit Runge Kutta method. See Sanderse [18].
    The implicit linear system is solved at each time step using Newton's method. The newton_type may be one of the following:
    • :no: Replace iteration matrix with I/Δt (no Jacobian)
    • :approximate: Build Jacobian once before iterations only
    • :full: Build Jacobian at each iteration
    Fields
    • A
    • b
    • c
    • r
    • newton_type
    • maxiter
    • abstol
    • reltol
    • p_add_solve
    source
    IncompressibleNavierStokes.RKMethods Module
    Set up Butcher arrays A, b, and c, as well as and SSP coefficient r. For families of methods, optional input s is the number of stages.
    Original (MATLAB) by David Ketcheson, extended by Benjamin Sanderse.
    Exports
    source
    IncompressibleNavierStokes.LMWray3 Type
    julia
    struct LMWray3{T} <: IncompressibleNavierStokes.AbstractRungeKuttaMethod{T}
    Low memory Wray 3rd order scheme. Uses 3 vector fields and one scalar field.
    Fields
    source
    Explicit Methods ​
    IncompressibleNavierStokes.RKMethods.FE11 Function
    julia
    FE11(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    FE11 (Forward Euler).
    source
    IncompressibleNavierStokes.RKMethods.SSP22 Function
    julia
    SSP22(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    FE11 (Forward Euler).
    source
    IncompressibleNavierStokes.RKMethods.SSP22 Function
    julia
    SSP22(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    SSP22.
    source
    IncompressibleNavierStokes.RKMethods.SSP42 Function
    julia
    SSP42(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    SSP22.
    source
    IncompressibleNavierStokes.RKMethods.SSP42 Function
    julia
    SSP42(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    SSP42.
    source
    IncompressibleNavierStokes.RKMethods.SSP33 Function
    julia
    SSP33(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    SSP42.
    source
    IncompressibleNavierStokes.RKMethods.SSP33 Function
    julia
    SSP33(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    SSP33.
    source
    IncompressibleNavierStokes.RKMethods.SSP43 Function
    julia
    SSP43(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    SSP33.
    source
    IncompressibleNavierStokes.RKMethods.SSP43 Function
    julia
    SSP43(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    SSP43.
    source
    IncompressibleNavierStokes.RKMethods.SSP104 Function
    julia
    SSP104(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    SSP43.
    source
    IncompressibleNavierStokes.RKMethods.SSP104 Function
    julia
    SSP104(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    SSP104.
    source
    IncompressibleNavierStokes.RKMethods.rSSPs2 Function
    julia
    rSSPs2(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    SSP104.
    source
    IncompressibleNavierStokes.RKMethods.rSSPs2 Function
    julia
    rSSPs2(
 ;
     ...
 ) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod, IncompressibleNavierStokes.ImplicitRungeKuttaMethod}
 rSSPs2(
     s;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod, IncompressibleNavierStokes.ImplicitRungeKuttaMethod}
    Rational (optimal, low-storage) s-stage 2nd order SSP.
    source
    IncompressibleNavierStokes.RKMethods.rSSPs3 Function
    julia
    rSSPs3(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod, IncompressibleNavierStokes.ImplicitRungeKuttaMethod}
    Rational (optimal, low-storage) s-stage 2nd order SSP.
    source
    IncompressibleNavierStokes.RKMethods.rSSPs3 Function
    julia
    rSSPs3(
 ;
     ...
 ) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod, IncompressibleNavierStokes.ImplicitRungeKuttaMethod}
 rSSPs3(
     s;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod, IncompressibleNavierStokes.ImplicitRungeKuttaMethod}
    Rational (optimal, low-storage) s^2-stage 3rd order SSP.
    source
    IncompressibleNavierStokes.RKMethods.Wray3 Function
    julia
    Wray3(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod, IncompressibleNavierStokes.ImplicitRungeKuttaMethod}
    Rational (optimal, low-storage) s^2-stage 3rd order SSP.
    source
    IncompressibleNavierStokes.RKMethods.Wray3 Function
    julia
    Wray3(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    Wray's RK3.
    source
    IncompressibleNavierStokes.RKMethods.RK56 Function
    julia
    RK56(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    Wray's RK3.
    source
    IncompressibleNavierStokes.RKMethods.RK56 Function
    julia
    RK56(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    RK56.
    source
    IncompressibleNavierStokes.RKMethods.DOPRI6 Function
    julia
    DOPRI6(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    RK56.
    source
    IncompressibleNavierStokes.RKMethods.DOPRI6 Function
    julia
    DOPRI6(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    Dormand-Price pair.
    source
    Implicit Methods ​
    IncompressibleNavierStokes.RKMethods.BE11 Function
    julia
    BE11(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    Dormand-Price pair.
    source
    Implicit Methods ​
    IncompressibleNavierStokes.RKMethods.BE11 Function
    julia
    BE11(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    Backward Euler.
    source
    IncompressibleNavierStokes.RKMethods.SDIRK34 Function
    julia
    SDIRK34(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    Backward Euler.
    source
    IncompressibleNavierStokes.RKMethods.SDIRK34 Function
    julia
    SDIRK34(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    3-stage, 4th order singly diagonally implicit (SSP).
    source
    IncompressibleNavierStokes.RKMethods.ISSPm2 Function
    julia
    ISSPm2(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    3-stage, 4th order singly diagonally implicit (SSP).
    source
    IncompressibleNavierStokes.RKMethods.ISSPm2 Function
    julia
    ISSPm2(
 ;
     ...
 ) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
 ISSPm2(
     s;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod, IncompressibleNavierStokes.ImplicitRungeKuttaMethod}
    Optimal DIRK SSP schemes of order 2.
    source
    IncompressibleNavierStokes.RKMethods.ISSPs3 Function
    julia
    ISSPs3(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod, IncompressibleNavierStokes.ImplicitRungeKuttaMethod}
    Optimal DIRK SSP schemes of order 2.
    source
    IncompressibleNavierStokes.RKMethods.ISSPs3 Function
    julia
    ISSPs3(
 ;
     ...
 ) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
 ISSPs3(
     s;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod, IncompressibleNavierStokes.ImplicitRungeKuttaMethod}
    Optimal DIRK SSP schemes of order 3.
    source
    Half explicit methods ​
    IncompressibleNavierStokes.RKMethods.HEM3 Function
    julia
    HEM3(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod, IncompressibleNavierStokes.ImplicitRungeKuttaMethod}
    Optimal DIRK SSP schemes of order 3.
    source
    Half explicit methods ​
    IncompressibleNavierStokes.RKMethods.HEM3 Function
    julia
    HEM3(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    Brasey and Hairer.
    source
    IncompressibleNavierStokes.RKMethods.HEM3BS Function
    julia
    HEM3BS(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    Brasey and Hairer.
    source
    IncompressibleNavierStokes.RKMethods.HEM3BS Function
    julia
    HEM3BS(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    HEM3BS.
    source
    IncompressibleNavierStokes.RKMethods.HEM5 Function
    julia
    HEM5(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    HEM3BS.
    source
    IncompressibleNavierStokes.RKMethods.HEM5 Function
    julia
    HEM5(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    Brasey and Hairer, 5 stage, 4th order.
    source
    Classical Methods ​
    IncompressibleNavierStokes.RKMethods.GL1 Function
    julia
    GL1(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    Brasey and Hairer, 5 stage, 4th order.
    source
    Classical Methods ​
    IncompressibleNavierStokes.RKMethods.GL1 Function
    julia
    GL1(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    GL1.
    source
    IncompressibleNavierStokes.RKMethods.GL2 Function
    julia
    GL2(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    GL1.
    source
    IncompressibleNavierStokes.RKMethods.GL2 Function
    julia
    GL2(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    GL2.
    source
    IncompressibleNavierStokes.RKMethods.GL3 Function
    julia
    GL3(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    GL2.
    source
    IncompressibleNavierStokes.RKMethods.GL3 Function
    julia
    GL3(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    GL3.
    source
    IncompressibleNavierStokes.RKMethods.RIA1 Function
    julia
    RIA1(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    GL3.
    source
    IncompressibleNavierStokes.RKMethods.RIA1 Function
    julia
    RIA1(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    This is implicit Euler.
    source
    IncompressibleNavierStokes.RKMethods.RIA2 Function
    julia
    RIA2(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    This is implicit Euler.
    source
    IncompressibleNavierStokes.RKMethods.RIA2 Function
    julia
    RIA2(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    RIA2.
    source
    IncompressibleNavierStokes.RKMethods.RIA3 Function
    julia
    RIA3(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    RIA2.
    source
    IncompressibleNavierStokes.RKMethods.RIA3 Function
    julia
    RIA3(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    RIA3.
    source
    IncompressibleNavierStokes.RKMethods.RIIA1 Function
    julia
    RIIA1(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    RIA3.
    source
    IncompressibleNavierStokes.RKMethods.RIIA1 Function
    julia
    RIIA1(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    RIIA1.
    source
    IncompressibleNavierStokes.RKMethods.RIIA2 Function
    julia
    RIIA2(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    RIIA1.
    source
    IncompressibleNavierStokes.RKMethods.RIIA2 Function
    julia
    RIIA2(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    RIIA2.
    source
    IncompressibleNavierStokes.RKMethods.RIIA3 Function
    julia
    RIIA3(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    RIIA2.
    source
    IncompressibleNavierStokes.RKMethods.RIIA3 Function
    julia
    RIIA3(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    RIIA3.
    source
    IncompressibleNavierStokes.RKMethods.LIIIA2 Function
    julia
    LIIIA2(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    RIIA3.
    source
    IncompressibleNavierStokes.RKMethods.LIIIA2 Function
    julia
    LIIIA2(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    LIIIA2.
    source
    IncompressibleNavierStokes.RKMethods.LIIIA3 Function
    julia
    LIIIA3(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    LIIIA2.
    source
    IncompressibleNavierStokes.RKMethods.LIIIA3 Function
    julia
    LIIIA3(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    LIIIA3.
    source
    Chebyshev methods ​
    IncompressibleNavierStokes.RKMethods.CHDIRK3 Function
    julia
    CHDIRK3(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    LIIIA3.
    source
    Chebyshev methods ​
    IncompressibleNavierStokes.RKMethods.CHDIRK3 Function
    julia
    CHDIRK3(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    Chebyshev based DIRK (not algebraically stable).
    source
    IncompressibleNavierStokes.RKMethods.CHCONS3 Function
    julia
    CHCONS3(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    Chebyshev based DIRK (not algebraically stable).
    source
    IncompressibleNavierStokes.RKMethods.CHCONS3 Function
    julia
    CHCONS3(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    CHCONS3.
    source
    IncompressibleNavierStokes.RKMethods.CHC3 Function
    julia
    CHC3(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    CHCONS3.
    source
    IncompressibleNavierStokes.RKMethods.CHC3 Function
    julia
    CHC3(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    Chebyshev quadrature and C(3) satisfied. Note this equals Lobatto IIIA.
    source
    IncompressibleNavierStokes.RKMethods.CHC5 Function
    julia
    CHC5(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    Chebyshev quadrature and C(3) satisfied. Note this equals Lobatto IIIA.
    source
    IncompressibleNavierStokes.RKMethods.CHC5 Function
    julia
    CHC5(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    CHC5.
    source
    Miscellaneous Methods ​
    IncompressibleNavierStokes.RKMethods.Mid22 Function
    julia
    Mid22(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    CHC5.
    source
    Miscellaneous Methods ​
    IncompressibleNavierStokes.RKMethods.Mid22 Function
    julia
    Mid22(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    Midpoint 22 method.
    source
    IncompressibleNavierStokes.RKMethods.MTE22 Function
    julia
    MTE22(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    Midpoint 22 method.
    source
    IncompressibleNavierStokes.RKMethods.MTE22 Function
    julia
    MTE22(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    Minimal truncation error 22 method (Heun).
    source
    IncompressibleNavierStokes.RKMethods.CN22 Function
    julia
    CN22(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    Minimal truncation error 22 method (Heun).
    source
    IncompressibleNavierStokes.RKMethods.CN22 Function
    julia
    CN22(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    Crank-Nicholson.
    source
    IncompressibleNavierStokes.RKMethods.Heun33 Function
    julia
    Heun33(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    Crank-Nicholson.
    source
    IncompressibleNavierStokes.RKMethods.Heun33 Function
    julia
    Heun33(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    Heun33.
    source
    IncompressibleNavierStokes.RKMethods.RK33C2 Function
    julia
    RK33C2(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    Heun33.
    source
    IncompressibleNavierStokes.RKMethods.RK33C2 Function
    julia
    RK33C2(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    RK3 satisfying C(2) for i=3.
    source
    IncompressibleNavierStokes.RKMethods.RK33P2 Function
    julia
    RK33P2(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    RK3 satisfying C(2) for i=3.
    source
    IncompressibleNavierStokes.RKMethods.RK33P2 Function
    julia
    RK33P2(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    RK3 satisfying the second order condition for the pressure.
    source
    IncompressibleNavierStokes.RKMethods.RK44 Function
    julia
    RK44(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    RK3 satisfying the second order condition for the pressure.
    source
    IncompressibleNavierStokes.RKMethods.RK44 Function
    julia
    RK44(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    Classical fourth order.
    source
    IncompressibleNavierStokes.RKMethods.RK44C2 Function
    julia
    RK44C2(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    Classical fourth order.
    source
    IncompressibleNavierStokes.RKMethods.RK44C2 Function
    julia
    RK44C2(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    RK4 satisfying C(2) for i=3.
    source
    IncompressibleNavierStokes.RKMethods.RK44C23 Function
    julia
    RK44C23(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    RK4 satisfying C(2) for i=3.
    source
    IncompressibleNavierStokes.RKMethods.RK44C23 Function
    julia
    RK44C23(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    RK4 satisfying C(2) for i=3 and c2=c3.
    source
    IncompressibleNavierStokes.RKMethods.RK44P2 Function
    julia
    RK44P2(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    RK4 satisfying C(2) for i=3 and c2=c3.
    source
    IncompressibleNavierStokes.RKMethods.RK44P2 Function
    julia
    RK44P2(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    RK4 satisfying the second order condition for the pressure (but not third order).
    source
    DSRK Methods ​
    IncompressibleNavierStokes.RKMethods.DSso2 Function
    julia
    DSso2(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    RK4 satisfying the second order condition for the pressure (but not third order).
    source
    DSRK Methods ​
    IncompressibleNavierStokes.RKMethods.DSso2 Function
    julia
    DSso2(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    CBM's DSRKso2.
    source
    IncompressibleNavierStokes.RKMethods.DSRK2 Function
    julia
    DSRK2(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    CBM's DSRKso2.
    source
    IncompressibleNavierStokes.RKMethods.DSRK2 Function
    julia
    DSRK2(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    CBM's DSRK2.
    source
    IncompressibleNavierStokes.RKMethods.DSRK3 Function
    julia
    DSRK3(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    CBM's DSRK2.
    source
    IncompressibleNavierStokes.RKMethods.DSRK3 Function
    julia
    DSRK3(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    Zennaro's DSRK3.
    source
    "Non-SSP" Methods of Wong & Spiteri ​
    IncompressibleNavierStokes.RKMethods.NSSP21 Function
    julia
    NSSP21(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    Zennaro's DSRK3.
    source
    "Non-SSP" Methods of Wong & Spiteri ​
    IncompressibleNavierStokes.RKMethods.NSSP21 Function
    julia
    NSSP21(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    NSSP21.
    source
    IncompressibleNavierStokes.RKMethods.NSSP32 Function
    julia
    NSSP32(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    NSSP21.
    source
    IncompressibleNavierStokes.RKMethods.NSSP32 Function
    julia
    NSSP32(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    NSSP32.
    source
    IncompressibleNavierStokes.RKMethods.NSSP33 Function
    julia
    NSSP33(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    NSSP32.
    source
    IncompressibleNavierStokes.RKMethods.NSSP33 Function
    julia
    NSSP33(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    NSSP33.
    source
    IncompressibleNavierStokes.RKMethods.NSSP53 Function
    julia
    NSSP53(
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    NSSP33.
    source
    IncompressibleNavierStokes.RKMethods.NSSP53 Function
    julia
    NSSP53(
 ;
     kwargs...
-) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}
    NSSP53.
    source
    
-    
+) -> Union{IncompressibleNavierStokes.ExplicitRungeKuttaMethod{Float64}, IncompressibleNavierStokes.ImplicitRungeKuttaMethod{Float64}}

    NSSP53.

    source

    + \ No newline at end of file diff --git a/dev/manual/utils.html b/dev/manual/utils.html index 119942981..d6c2e8b2a 100644 --- a/dev/manual/utils.html +++ b/dev/manual/utils.html @@ -9,11 +9,11 @@ - + - + - + @@ -24,17 +24,17 @@
    IncompressibleNavierStokes.jl

    Appearance

    Utils ​

    IncompressibleNavierStokes.get_lims Function
    julia
    get_lims(x) -> Tuple{Any, Any}
-get_lims(x, n) -> Tuple{Any, Any}

    Get approximate lower and upper limits of a field x based on the mean and standard deviation (μ±nσ). If x is constant, a margin of 1e-4 is enforced. This is required for contour plotting functions that require a certain range.

    source

    IncompressibleNavierStokes.get_spectrum Method
    julia
    get_spectrum(
+get_lims(x, n) -> Tuple{Any, Any}

    Get approximate lower and upper limits of a field x based on the mean and standard deviation (μ±nσ). If x is constant, a margin of 1e-4 is enforced. This is required for contour plotting functions that require a certain range.

    source

    IncompressibleNavierStokes.get_spectrum Method
    julia
    get_spectrum(
     setup;
     npoint,
     a
-) -> NamedTuple{(:κ, :masks, :K), <:Tuple{Any, Any, Any}}

    Get energy spectrum of velocity field.

    source

    IncompressibleNavierStokes.getoffset Method
    julia
    getoffset(I) -> Any

    Get offset from CartesianIndices I.

    source

    IncompressibleNavierStokes.getval Method
    julia
    getval(_::Val{x}) -> Any

    Get value contained in Val.

    source

    IncompressibleNavierStokes.plotgrid Function
    julia
    plotgrid(x, y; kwargs...)
-plotgrid(x, y, z)

    Plot nonuniform Cartesian grid.

    source

    IncompressibleNavierStokes.spectral_stuff Method
    julia
    spectral_stuff(
+) -> NamedTuple{(:κ, :masks, :K), <:Tuple{Any, Any, Any}}

    Get energy spectrum of velocity field.

    source

    IncompressibleNavierStokes.getoffset Method
    julia
    getoffset(I) -> Any

    Get offset from CartesianIndices I.

    source

    IncompressibleNavierStokes.getval Method
    julia
    getval(_::Val{x}) -> Any

    Get value contained in Val.

    source

    IncompressibleNavierStokes.plotgrid Function
    julia
    plotgrid(x, y; kwargs...)
+plotgrid(x, y, z)

    Plot nonuniform Cartesian grid.

    source

    IncompressibleNavierStokes.spectral_stuff Method
    julia
    spectral_stuff(
     setup;
     npoint,
     a
-) -> NamedTuple{(:inds, :κ, :K), <:Tuple{Any, Any, Any}}

    Get utilities to compute energy spectrum.

    source

    IncompressibleNavierStokes.splitseed Method
    julia
    splitseed(seed, n) -> AbstractArray

    Split random number generator seed into n new seeds.

    source

    - +) -> NamedTuple{(:inds, :κ, :K), <:Tuple{Any, Any, Any}}

    Get utilities to compute energy spectrum.

    source

    IncompressibleNavierStokes.splitseed Method
    julia
    splitseed(seed, n) -> AbstractArray

    Split random number generator seed into n new seeds.

    source

    + \ No newline at end of file diff --git a/dev/references.html b/dev/references.html index 30023bbb0..ac861cfcf 100644 --- a/dev/references.html +++ b/dev/references.html @@ -9,9 +9,9 @@ - + - + @@ -24,7 +24,7 @@
    IncompressibleNavierStokes.jl

    Appearance

    References ​

    Bibliography ​

    1. D. Kochkov, J. A. Smith, A. Alieva, Q. Wang, M. P. Brenner and S. Hoyer. Machine learning-accelerated computational fluid dynamics. Proceedings of the National Academy of Sciences 118 (2021).

    2. M. Kurz, P. Offenhäuser and A. Beck. Deep Reinforcement Learning for Turbulence Modeling in Large Eddy Simulations, arXiv (2022).

    3. B. List, L.-W. Chen and N. Thuerey. Learned Turbulence Modelling with Differentiable Fluid Solvers, arxiv:2202.06988 (2022).

    4. J. F. MacArt, J. Sirignano and J. B. Freund. Embedded training of neural-network sub-grid-scale turbulence models (2021).

    5. J. Li and P. M. Carrica. A simple approach for vortex core visualization, arXiv:1910.06998 (2019), arXiv:1910.06998 [physics.flu-dyn].

    6. J. Jeong and F. Hussain. On the identification of a vortex. J. Fluid. Mech. 285, 69–94 (1995).

    7. S. B. Pope. Turbulent Flows (Cambridge University Press, Cambridge, England, 2000).

    8. B. Sanderse and F. X. Trias. Energy-consistent discretization of viscous dissipation with application to natural convection flow (Jul 2023), arXiv:2307.10874v1 [physics.flu-dyn].

    9. M. H. Silvis, R. A. Remmerswaal and R. Verstappen. Physical consistency of subgrid-scale models for large-eddy simulation of incompressible turbulent flows. Physics of Fluids 29 (2017).

    10. F. X. TRIAS, M. SORIA, A. OLIVA and C. D. PÉREZ-SEGARRA. Direct numerical simulations of two- and three-dimensional turbulent natural convection flows in a differentially heated cavity of aspect ratio 4. Journal of Fluid Mechanics 586, 259–293 (2007).

    11. P. Orlandi. Fluid flow phenomena: a numerical toolkit. Vol. 55 (Springer Science & Business Media, 2000).

    12. O. San and A. E. Staples. High-order methods for decaying two-dimensional homogeneous isotropic turbulence. Computers & Fluids 63, 105–127 (2012).

    13. F. H. Harlow and J. E. Welch. * Numerical Calculation of Time‐Dependent Viscous Incompressible Flow of Fluid with Free Surface *. The Physics of Fluids 8, 2182–2189 (1965), arXiv:https://aip.scitation.org/doi/pdf/10.1063/1.1761178.

    14. R. W. Verstappen and A. E. Veldman. Symmetry-Preserving Discretization of Turbulent Flow. J. Comput. Phys. 187, 343–368 (2003).

    15. B. Sanderse, R. Verstappen and B. Koren. Boundary treatment for fourth-order staggered mesh discretizations of the incompressible Navier–Stokes equations. Journal of Computational Physics 257, 1472–1505 (2014). Physics-compatible numerical methods.

    16. R. Verstappen and A. Veldman. Direct Numerical Simulation of Turbulence at Lower Costs. Journal of Engineering Mathematics 32, 143–159 (1997).

    17. B. Sanderse and B. Koren. Accuracy analysis of explicit Runge–Kutta methods applied to the incompressible Navier–Stokes equations. Journal of Computational Physics 231, 3041–3063 (2012).

    18. B. Sanderse. Energy-conserving Runge–Kutta methods for the incompressible Navier–Stokes equations. Journal of Computational Physics 233, 100–131 (2013).

    - + \ No newline at end of file