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<!DOCTYPE html>
<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>ConservationLaws · StableSpectralElements.jl</title><script data-outdated-warner src="../assets/warner.js"></script><link href="https://cdnjs.cloudflare.com/ajax/libs/lato-font/3.0.0/css/lato-font.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/juliamono/0.045/juliamono.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.4/css/fontawesome.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.4/css/solid.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.4/css/brands.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.24/katex.min.css" rel="stylesheet" type="text/css"/><script>documenterBaseURL=".."</script><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.3.6/require.min.js" data-main="../assets/documenter.js"></script><script src="../siteinfo.js"></script><script src="../../versions.js"></script><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/documenter-dark.css" data-theme-name="documenter-dark" data-theme-primary-dark/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/documenter-light.css" data-theme-name="documenter-light" data-theme-primary/><script src="../assets/themeswap.js"></script><link href="../assets/favicon.ico" rel="icon" type="image/x-icon"/></head><body><div id="documenter"><nav class="docs-sidebar"><div class="docs-package-name"><span class="docs-autofit"><a href="../">StableSpectralElements.jl</a></span></div><form class="docs-search" action="../search/"><input class="docs-search-query" id="documenter-search-query" name="q" type="text" placeholder="Search docs"/></form><ul class="docs-menu"><li><a class="tocitem" href="../">Home</a></li><li><span class="tocitem">Reference</span><ul><li class="is-active"><a class="tocitem" href><code>ConservationLaws</code></a><ul class="internal"><li><a class="tocitem" href="#Overview"><span>Overview</span></a></li><li><a class="tocitem" href="#Equations"><span>Equations</span></a></li></ul></li><li><a class="tocitem" href="../SpatialDiscretizations/"><code>SpatialDiscretizations</code></a></li></ul></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Reference</a></li><li class="is-active"><a href><code>ConservationLaws</code></a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href><code>ConservationLaws</code></a></li></ul></nav><div class="docs-right"><a class="docs-edit-link" href="https://github.com/tristanmontoya/StableSpectralElements.jl/blob/main/docs/src/ConservationLaws.md" title="Edit on GitHub"><span class="docs-icon fab"></span><span class="docs-label is-hidden-touch">Edit on GitHub</span></a><a class="docs-settings-button fas fa-cog" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-sidebar-button fa fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a></div></header><article class="content" id="documenter-page"><h1 id="Module-ConservationLaws"><a class="docs-heading-anchor" href="#Module-ConservationLaws">Module <code>ConservationLaws</code></a><a id="Module-ConservationLaws-1"></a><a class="docs-heading-anchor-permalink" href="#Module-ConservationLaws" title="Permalink"></a></h1><h2 id="Overview"><a class="docs-heading-anchor" href="#Overview">Overview</a><a id="Overview-1"></a><a class="docs-heading-anchor-permalink" href="#Overview" title="Permalink"></a></h2><p>The equations to be solved are defined by subtypes of <code>AbstractConservationLaw</code> on which functions such as <code>physical_flux</code> and <code>numerical_flux</code> are dispatched. Objects of type <code>AbstractConservationLaw</code> contain two type parameters, <code>d</code> and <code>PDEType</code>, the former denoting the spatial dimension of the problem, which is inherited by all subtypes, and the latter being a subtype of <code>AbstractPDEType</code> denoting the particular type of PDE being solved, which is either <code>FirstOrder</code> or <code>SecondOrder</code>. Whereas first-order problems remove the dependence of the flux tensor on the solution gradient in order to obtain systems of the form</p><p class="math-container">\[\partial_t \underline{U}(\bm{x},t) + \bm{\nabla}_{\bm{x}} \cdot \underline{\bm{F}}(\underline{U}(\bm{x},t)) = \underline{0},\]</p><p>second-order problems are treated by StableSpectralElements.jl as first-order systems of the form </p><p class="math-container">\[\begin{aligned}
\underline{\bm{Q}}(\bm{x},t) - \bm{\nabla}_{\bm{x}} \underline{U}(\bm{x},t) &amp;= \underline{0},\\
\partial_t \underline{U}(\bm{x},t) + \bm{\nabla}_{\bm{x}} \cdot \underline{\bm{F}}(\underline{U}(\bm{x},t), \underline{\bm{Q}}(\bm{x},t)) &amp;= \underline{0}.
\end{aligned}\]</p><p>Currently, the linear advection and advection-diffusion equations, the inviscid and viscous Burgers&#39; 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.</p><h2 id="Equations"><a class="docs-heading-anchor" href="#Equations">Equations</a><a id="Equations-1"></a><a class="docs-heading-anchor-permalink" href="#Equations" title="Permalink"></a></h2><p>Listed below are partial differential equations supported by StableSpectralElements.jl.</p><article class="docstring"><header><a class="docstring-binding" id="StableSpectralElements.ConservationLaws.LinearAdvectionEquation" href="#StableSpectralElements.ConservationLaws.LinearAdvectionEquation"><code>StableSpectralElements.ConservationLaws.LinearAdvectionEquation</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">LinearAdvectionEquation(a::NTuple{d,Float64}) where {d}</code></pre><p>Define a linear advection equation of the form</p><p class="math-container">\[\partial_t U(\bm{x},t) + \bm{\nabla} \cdot \big( \bm{a} U(\bm{x},t) \big) = 0,\]</p><p>with a constant advection velocity <span>$\bm{a} \in \R^d$</span>. A specialized constructor <code>LinearAdvectionEquation(a::Float64)</code> is provided for the one-dimensional case.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/tristanmontoya/StableSpectralElements.jl/blob/b65332361cae85f770ba4c06ae0ab458fbf5124d/src/ConservationLaws/linear_advection_diffusion.jl#L1-L9">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="StableSpectralElements.ConservationLaws.LinearAdvectionDiffusionEquation" href="#StableSpectralElements.ConservationLaws.LinearAdvectionDiffusionEquation"><code>StableSpectralElements.ConservationLaws.LinearAdvectionDiffusionEquation</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">LinearAdvectionDiffusionEquation(a::NTuple{d,Float64}, b::Float64) where {d}</code></pre><p>Define a linear advection-diffusion equation of the form</p><p class="math-container">\[\partial_t U(\bm{x},t) + \bm{\nabla} \cdot \big( \bm{a} U(\bm{x},t) - b \bm{\nabla} U(\bm{x},t)\big) = 0,\]</p><p>with a constant advection velocity <span>$\bm{a} \in \R^d$</span> and diffusion coefficient <span>$b \in \R^+$</span>. A specialized constructor <code>LinearAdvectionDiffusionEquation(a::Float64, b::Float64)</code> is provided for the one-dimensional case.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/tristanmontoya/StableSpectralElements.jl/blob/b65332361cae85f770ba4c06ae0ab458fbf5124d/src/ConservationLaws/linear_advection_diffusion.jl#L21-L29">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="StableSpectralElements.ConservationLaws.InviscidBurgersEquation" href="#StableSpectralElements.ConservationLaws.InviscidBurgersEquation"><code>StableSpectralElements.ConservationLaws.InviscidBurgersEquation</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">InviscidBurgersEquation(a::NTuple{d,Float64}) where {d}</code></pre><p>Define an inviscid Burgers&#39; equation of the form</p><p class="math-container">\[\partial_t U(\bm{x},t) + \bm{\nabla} \cdot \big(\tfrac{1}{2}\bm{a} U(\bm{x},t)^2 \big) = 0,\]</p><p>where <span>$\bm{a} \in \R^d$</span>. A specialized constructor <code>InviscidBurgersEquation()</code> is provided for the one-dimensional case with <code>a = (1.0,)</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/tristanmontoya/StableSpectralElements.jl/blob/b65332361cae85f770ba4c06ae0ab458fbf5124d/src/ConservationLaws/burgers.jl#L1-L9">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="StableSpectralElements.ConservationLaws.ViscousBurgersEquation" href="#StableSpectralElements.ConservationLaws.ViscousBurgersEquation"><code>StableSpectralElements.ConservationLaws.ViscousBurgersEquation</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ViscousBurgersEquation(a::NTuple{d,Float64}, b::Float64) where {d}</code></pre><p>Define a viscous Burgers&#39; equation of the form</p><p class="math-container">\[\partial_t U(\bm{x},t) + \bm{\nabla} \cdot \big(\tfrac{1}{2}\bm{a} U(\bm{x},t)^2 - b \bm{\nabla} U(\bm{x},t)\big) = 0,\]</p><p>where <span>$\bm{a} \in \R^d$</span> and <span>$b \in \R^+$</span>. A specialized constructor <code>ViscousBurgersEquation(b::Float64)</code> is provided for the one-dimensional case with <code>a = (1.0,)</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/tristanmontoya/StableSpectralElements.jl/blob/b65332361cae85f770ba4c06ae0ab458fbf5124d/src/ConservationLaws/burgers.jl#L21-L29">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="StableSpectralElements.ConservationLaws.EulerEquations" href="#StableSpectralElements.ConservationLaws.EulerEquations"><code>StableSpectralElements.ConservationLaws.EulerEquations</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">EulerEquations{d}(γ::Float64) where {d}</code></pre><p>Define an Euler system governing compressible, adiabatic fluid flow, taking the form</p><p class="math-container">\[\frac{\partial}{\partial t}\left[\begin{array}{c}
\rho(\bm{x}, t) \\
\rho(\bm{x}, t) V_1(\bm{x}, t) \\
\vdots \\
\rho(\bm{x}, t) V_d(\bm{x}, t) \\
E(\bm{x}, t)
\end{array}\right]+\sum_{m=1}^d \frac{\partial}{\partial x_m}\left[\begin{array}{c}
\rho(\bm{x}, t) V_m(\bm{x}, t) \\
\rho(\bm{x}, t) V_1(\bm{x}, t) V_m(\bm{x}, t)+P(\bm{x}, t) \delta_{1 m} \\
\vdots \\
\rho(\bm{x}, t) V_d(\bm{x}, t) V_m(\bm{x}, t)+P(\bm{x}, t) \delta_{d m} \\
V_m(\bm{x}, t)(E(\bm{x}, t)+P(\bm{x}, t))
\end{array}\right]=\underline{0},\]</p><p>where <span>$\rho(\bm{x},t) \in \mathbb{R}$</span> is the fluid density, <span>$\bm{V}(\bm{x},t) \in \mathbb{R}^d$</span> is the flow velocity, <span>$E(\bm{x},t) \in \mathbb{R}$</span> is the total energy per unit volume, and the pressure is given for an ideal gas with constant specific heat as</p><p class="math-container">\[P(\bm{x},t) = (\gamma - 1)\Big(E(\bm{x},t) - \frac{1}{2}\rho(\bm{x},t) \lVert \bm{V}(\bm{x},t)\rVert^2\Big).\]</p><p>The specific heat ratio is specified as a parameter <code>γ::Float64</code>, which must be greater than unity.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/tristanmontoya/StableSpectralElements.jl/blob/b65332361cae85f770ba4c06ae0ab458fbf5124d/src/ConservationLaws/euler_navierstokes.jl#L1-L25">source</a></section></article></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../">« Home</a><a class="docs-footer-nextpage" href="../SpatialDiscretizations/"><code>SpatialDiscretizations</code> »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 0.27.25 on <span class="colophon-date" title="Wednesday 2 August 2023 21:41">Wednesday 2 August 2023</span>. Using Julia version 1.8.5.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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