From 9a08e0c054dba691908abc78bb10ab287ff05364 Mon Sep 17 00:00:00 2001 From: "Documenter.jl" Date: Mon, 31 Jul 2023 19:39:32 +0000 Subject: [PATCH] build based on 9b70100 --- dev/ConservationLaws/index.html | 4 ++-- dev/SpatialDiscretizations/index.html | 2 +- dev/index.html | 2 +- dev/search/index.html | 2 +- 4 files changed, 5 insertions(+), 5 deletions(-) diff --git a/dev/ConservationLaws/index.html b/dev/ConservationLaws/index.html index 66efc2f9..2e2f4477 100644 --- a/dev/ConservationLaws/index.html +++ b/dev/ConservationLaws/index.html @@ -2,7 +2,7 @@ ConservationLaws · StableSpectralElements.jl
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Module ConservationLaws

Overview

The equations to be solved are defined by subtypes of AbstractConservationLaw on which functions such as physical_flux and numerical_flux are dispatched. Objects of type AbstractConservationLaw contain two type parameters, d and PDEType, the former denoting the spatial dimension of the problem, which is inherited by all subtypes, and the latter being a subtype of AbstractPDEType denoting the particular type of PDE being solved, which is either FirstOrder or SecondOrder. Whereas first-order problems remove the dependence of the flux tensor on the solution gradient in order to obtain systems of the form

\[\partial_t \underline{U}(\bm{x},t) + \bm{\nabla}_{\bm{x}} \cdot \underline{\bm{F}}(\underline{U}(\bm{x},t)) = \underline{0},\]

second-order problems are treated by StableSpectralElements.jl as first-order systems of the form

\[\begin{aligned} \underline{\bm{Q}}(\bm{x},t) - \bm{\nabla}_{\bm{x}} \underline{U}(\bm{x},t) &= \underline{0},\\ \partial_t \underline{U}(\bm{x},t) + \bm{\nabla}_{\bm{x}} \cdot \underline{\bm{F}}(\underline{U}(\bm{x},t), \underline{\bm{Q}}(\bm{x},t)) &= \underline{0}. -\end{aligned}\]

Currently, the linear advection and advection-diffusion equations, the inviscid and viscous Burgers' equations, and the compressible Euler equations are supported by StableSpectralElements.jl, but any system of the above form can in principle be implemented, provided that appropriate physical and numerical fluxes are defined.

Equations

Listed below are partial differential equations supported by StableSpectralElements.jl.

StableSpectralElements.ConservationLaws.LinearAdvectionEquation — Type
LinearAdvectionEquation(a::NTuple{d,Float64}) where {d}

Define a linear advection equation of the form

\[\partial_t U(\bm{x},t) + \bm{\nabla} \cdot \big( \bm{a} U(\bm{x},t) \big) = 0,\]

with a constant advection velocity $\bm{a} \in \R^d$. A specialized constructor LinearAdvectionEquation(a::Float64) is provided for the one-dimensional case.

source
StableSpectralElements.ConservationLaws.LinearAdvectionDiffusionEquation — Type
LinearAdvectionDiffusionEquation(a::NTuple{d,Float64}, b::Float64) where {d}

Define a linear advection-diffusion equation of the form

\[\partial_t U(\bm{x},t) + \bm{\nabla} \cdot \big( \bm{a} U(\bm{x},t) - b \bm{\nabla} U(\bm{x},t)\big) = 0,\]

with a constant advection velocity $\bm{a} \in \R^d$ and diffusion coefficient $b \in \R^+$. A specialized constructor LinearAdvectionDiffusionEquation(a::Float64, b::Float64) is provided for the one-dimensional case.

source
StableSpectralElements.ConservationLaws.InviscidBurgersEquation — Type
InviscidBurgersEquation(a::NTuple{d,Float64}) where {d}

Define an inviscid Burgers' equation of the form

\[\partial_t U(\bm{x},t) + \bm{\nabla} \cdot \big(\tfrac{1}{2}\bm{a} U(\bm{x},t)^2 \big) = 0,\]

where $\bm{a} \in \R^d$. A specialized constructor InviscidBurgersEquation() is provided for the one-dimensional case with a = (1.0,).

source
StableSpectralElements.ConservationLaws.ViscousBurgersEquation — Type
ViscousBurgersEquation(a::NTuple{d,Float64}, b::Float64) where {d}

Define a viscous Burgers' equation of the form

\[\partial_t U(\bm{x},t) + \bm{\nabla} \cdot \big(\tfrac{1}{2}\bm{a} U(\bm{x},t)^2 - b \bm{\nabla} U(\bm{x},t)\big) = 0,\]

where $\bm{a} \in \R^d$ and $b \in \R^+$. A specialized constructor ViscousBurgersEquation(b::Float64) is provided for the one-dimensional case with a = (1.0,).

source
StableSpectralElements.ConservationLaws.EulerEquations — Type
EulerEquations{d}(γ::Float64) where {d}

Define an Euler system governing compressible, adiabatic fluid flow, taking the form

\[\frac{\partial}{\partial t}\left[\begin{array}{c} +\end{aligned}\]

Currently, the linear advection and advection-diffusion equations, the inviscid and viscous Burgers' equations, and the compressible Euler equations are supported by StableSpectralElements.jl, but any system of the above form can in principle be implemented, provided that appropriate physical and numerical fluxes are defined.

Equations

Listed below are partial differential equations supported by StableSpectralElements.jl.

StableSpectralElements.ConservationLaws.LinearAdvectionEquation — Type
LinearAdvectionEquation(a::NTuple{d,Float64}) where {d}

Define a linear advection equation of the form

\[\partial_t U(\bm{x},t) + \bm{\nabla} \cdot \big( \bm{a} U(\bm{x},t) \big) = 0,\]

with a constant advection velocity $\bm{a} \in \R^d$. A specialized constructor LinearAdvectionEquation(a::Float64) is provided for the one-dimensional case.

source
StableSpectralElements.ConservationLaws.LinearAdvectionDiffusionEquation — Type
LinearAdvectionDiffusionEquation(a::NTuple{d,Float64}, b::Float64) where {d}

Define a linear advection-diffusion equation of the form

\[\partial_t U(\bm{x},t) + \bm{\nabla} \cdot \big( \bm{a} U(\bm{x},t) - b \bm{\nabla} U(\bm{x},t)\big) = 0,\]

with a constant advection velocity $\bm{a} \in \R^d$ and diffusion coefficient $b \in \R^+$. A specialized constructor LinearAdvectionDiffusionEquation(a::Float64, b::Float64) is provided for the one-dimensional case.

source
StableSpectralElements.ConservationLaws.InviscidBurgersEquation — Type
InviscidBurgersEquation(a::NTuple{d,Float64}) where {d}

Define an inviscid Burgers' equation of the form

\[\partial_t U(\bm{x},t) + \bm{\nabla} \cdot \big(\tfrac{1}{2}\bm{a} U(\bm{x},t)^2 \big) = 0,\]

where $\bm{a} \in \R^d$. A specialized constructor InviscidBurgersEquation() is provided for the one-dimensional case with a = (1.0,).

source
StableSpectralElements.ConservationLaws.ViscousBurgersEquation — Type
ViscousBurgersEquation(a::NTuple{d,Float64}, b::Float64) where {d}

Define a viscous Burgers' equation of the form

\[\partial_t U(\bm{x},t) + \bm{\nabla} \cdot \big(\tfrac{1}{2}\bm{a} U(\bm{x},t)^2 - b \bm{\nabla} U(\bm{x},t)\big) = 0,\]

where $\bm{a} \in \R^d$ and $b \in \R^+$. A specialized constructor ViscousBurgersEquation(b::Float64) is provided for the one-dimensional case with a = (1.0,).

source
StableSpectralElements.ConservationLaws.EulerEquations — Type
EulerEquations{d}(γ::Float64) where {d}

Define an Euler system governing compressible, adiabatic fluid flow, taking the form

\[\frac{\partial}{\partial t}\left[\begin{array}{c} \rho(\bm{x}, t) \\ \rho(\bm{x}, t) V_1(\bm{x}, t) \\ \vdots \\ @@ -14,4 +14,4 @@ \vdots \\ \rho(\bm{x}, t) V_d(\bm{x}, t) V_m(\bm{x}, t)+P(\bm{x}, t) \delta_{d m} \\ V_m(\bm{x}, t)(E(\bm{x}, t)+P(\bm{x}, t)) -\end{array}\right]=\underline{0},\]

where $\rho(\bm{x},t) \in \mathbb{R}$ is the fluid density, $\bm{V}(\bm{x},t) \in \mathbb{R}^d$ is the flow velocity, $E(\bm{x},t) \in \mathbb{R}$ is the total energy per unit volume, and the pressure is given for an ideal gas with constant specific heat as

\[P(\bm{x},t) = (\gamma - 1)\Big(E(\bm{x},t) - \frac{1}{2}\rho(\bm{x},t) \lVert \bm{V}(\bm{x},t)\rVert^2\Big).\]

The specific heat ratio is specified as a parameter γ::Float64, which must be greater than unity.

source
+\end{array}\right]=\underline{0},\]

where $\rho(\bm{x},t) \in \mathbb{R}$ is the fluid density, $\bm{V}(\bm{x},t) \in \mathbb{R}^d$ is the flow velocity, $E(\bm{x},t) \in \mathbb{R}$ is the total energy per unit volume, and the pressure is given for an ideal gas with constant specific heat as

\[P(\bm{x},t) = (\gamma - 1)\Big(E(\bm{x},t) - \frac{1}{2}\rho(\bm{x},t) \lVert \bm{V}(\bm{x},t)\rVert^2\Big).\]

The specific heat ratio is specified as a parameter γ::Float64, which must be greater than unity.

source diff --git a/dev/SpatialDiscretizations/index.html b/dev/SpatialDiscretizations/index.html index 4d44c444..4931546a 100644 --- a/dev/SpatialDiscretizations/index.html +++ b/dev/SpatialDiscretizations/index.html @@ -5,4 +5,4 @@ \hat{\Omega}_{\mathrm{hex}} & = [-1,1]^3, \\ \hat{\Omega}_{\mathrm{tri}} &= \big\{ \bm{\xi} \in [-1,1]^2 : \xi_1 + \xi_2 \leq 0 \big\},\\ \hat{\Omega}_{\mathrm{tet}} &= \big\{ \bm{\xi} \in [-1,1]^3 : \xi_1 + \xi_2 + \xi_3 \leq -1 \big\}. -\end{aligned}\]

These element types are used in the constructor for StableSpectralElements.jl's ReferenceApproximation type, along with a subtype of AbstractApproximationType specifying the nature of the local approximation (and, optionally, the associated volume and facet quadrature rules).

All the information used to define the spatial discretization on the physical domain $\Omega$ is contained within a SpatialDiscretization structure, which is constructed using a ReferenceApproximation and a MeshData from StartUpDG.jl, which are stored as the fields reference_approximation and mesh. When the constructor for a SpatialDiscretization is called, the grid metrics are computed and stored in a GeometricFactors structure, with the corresponding field being geometric_factors.

+\end{aligned}\]

These element types are used in the constructor for StableSpectralElements.jl's ReferenceApproximation type, along with a subtype of AbstractApproximationType specifying the nature of the local approximation (and, optionally, the associated volume and facet quadrature rules).

All the information used to define the spatial discretization on the physical domain $\Omega$ is contained within a SpatialDiscretization structure, which is constructed using a ReferenceApproximation and a MeshData from StartUpDG.jl, which are stored as the fields reference_approximation and mesh. When the constructor for a SpatialDiscretization is called, the grid metrics are computed and stored in a GeometricFactors structure, with the corresponding field being geometric_factors.

diff --git a/dev/index.html b/dev/index.html index 7f80137a..7687e8b2 100644 --- a/dev/index.html +++ b/dev/index.html @@ -1,2 +1,2 @@ -Home · StableSpectralElements.jl
Edit on GitHub

StableSpectralElements.jl

StableSpectralElements.jl, formerly known as CLOUD.jl (Conservation Laws on Unstructured Domains), is a Julia framework for the numerical solution of partial differential equations of the form

\[\partial_t \underline{U}(\bm{x},t) + \bm{\nabla}_{\bm{x}} \cdot \underline{\bm{F}}(\underline{U}(\bm{x},t), \bm{\nabla}_{\bm{x}}\underline{U}(\bm{x},t)) = \underline{0},\]

for $t \in (0,T)$ with $T \in \mathbb{R}^+ $ and $\bm{x} \in \Omega \subset \mathbb{R}^d$, subject to appropriate initial and boundary conditions, where $\underline{U}(\bm{x},t)$ is the vector of solution variables and $\underline{\bm{F}}(\underline{U}(\bm{x},t),\bm{\nabla}_{\bm{x}}\underline{U}(\bm{x},t))$ is the flux tensor containing advective and/or diffusive contributions. These equations are spatially discretized on curvilinear unstructured grids using energy-stable and entropy-stable discontinuous spectral element methods in order to generate ODEProblem objects suitable for time integration using OrdinaryDiffEq.jl within the SciML ecosystem. StableSpectralElements.jl also includes postprocessing tools employing WriteVTK.jl for generating .vtu files, allowing for visualization of high-order numerical solutions on unstructured grids using ParaView or other software. Shared-memory parallelization is supported through multithreading.

The functionality provided by StartUpDG.jl for the handling of mesh data structures, polynomial basis functions, and quadrature nodes is employed throughout this package. Moreover, StableSpectralElements.jl implements dispatched strategies for semi-discrete operator evaluation using LinearMaps.jl, allowing for the efficient matrix-free application of tensor-product operators.

Discretizations satisfying the summation-by-parts property employing nodal as well as modal bases are implemented, with the latter allowing for efficient and low-storage inversion of the dense elemental mass matrices arising from curvilinear meshes through the use of weight-adjusted approximations.

Installation

StableSpectralElements.jl is a registered Julia package (compatible with Julia versions 1.8 and higher) and can be installed by entering the following commands within the REPL:

using Pkg; Pkg.add("StableSpectralElements")

Examples

We recommend that users refer to the following Jupyter notebooks (included in the examples directory) for examples of how to use StableSpectralElements.jl:

License

This software is released under the GPLv3 license.

+Home · StableSpectralElements.jl
Edit on GitHub

StableSpectralElements.jl

StableSpectralElements.jl, formerly known as CLOUD.jl (Conservation Laws on Unstructured Domains), is a Julia framework for the numerical solution of partial differential equations of the form

\[\partial_t \underline{U}(\bm{x},t) + \bm{\nabla}_{\bm{x}} \cdot \underline{\bm{F}}(\underline{U}(\bm{x},t), \bm{\nabla}_{\bm{x}}\underline{U}(\bm{x},t)) = \underline{0},\]

for $t \in (0,T)$ with $T \in \mathbb{R}^+ $ and $\bm{x} \in \Omega \subset \mathbb{R}^d$, subject to appropriate initial and boundary conditions, where $\underline{U}(\bm{x},t)$ is the vector of solution variables and $\underline{\bm{F}}(\underline{U}(\bm{x},t),\bm{\nabla}_{\bm{x}}\underline{U}(\bm{x},t))$ is the flux tensor containing advective and/or diffusive contributions. These equations are spatially discretized on curvilinear unstructured grids using energy-stable and entropy-stable discontinuous spectral element methods in order to generate ODEProblem objects suitable for time integration using OrdinaryDiffEq.jl within the SciML ecosystem. StableSpectralElements.jl also includes postprocessing tools employing WriteVTK.jl for generating .vtu files, allowing for visualization of high-order numerical solutions on unstructured grids using ParaView or other software. Shared-memory parallelization is supported through multithreading.

The functionality provided by StartUpDG.jl for the handling of mesh data structures, polynomial basis functions, and quadrature nodes is employed throughout this package. Moreover, StableSpectralElements.jl implements dispatched strategies for semi-discrete operator evaluation using LinearMaps.jl, allowing for the efficient matrix-free application of tensor-product operators.

Discretizations satisfying the summation-by-parts property employing nodal as well as modal bases are implemented, with the latter allowing for efficient and low-storage inversion of the dense elemental mass matrices arising from curvilinear meshes through the use of weight-adjusted approximations.

Installation

StableSpectralElements.jl is a registered Julia package (compatible with Julia versions 1.8 and higher) and can be installed by entering the following commands within the REPL:

using Pkg; Pkg.add("StableSpectralElements")

Examples

We recommend that users refer to the following Jupyter notebooks (included in the examples directory) for examples of how to use StableSpectralElements.jl:

License

This software is released under the GPLv3 license.

diff --git a/dev/search/index.html b/dev/search/index.html index 7be1ccdd..2007911b 100644 --- a/dev/search/index.html +++ b/dev/search/index.html @@ -1,2 +1,2 @@ -Search · StableSpectralElements.jl

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