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<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script data-outdated-warner src="../assets/warner.js"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">\[\begin{aligned} | ||
\underline{\bm{Q}}(\bm{x},t) - \bm{\nabla}_{\bm{x}} \underline{U}(\bm{x},t) &= \underline{0},\\ | ||
\partial_t \underline{U}(\bm{x},t) + \bm{\nabla}_{\bm{x}} \cdot \underline{\bm{F}}(\underline{U}(\bm{x},t), \underline{\bm{Q}}(\bm{x},t)) &= \underline{0}. | ||
\end{aligned}\]</p><p>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.</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/09901cfc05d7da03b84e526e1866276f684340fa/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/09901cfc05d7da03b84e526e1866276f684340fa/src/ConservationLaws/linear_advection_diffusion.jl#L19-L27">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' 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/09901cfc05d7da03b84e526e1866276f684340fa/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><p>x ViscousBurgersEquation(a::NTuple{d,Float64}, b::Float64) where {d}</p><p>Define a viscous Burgers' 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/09901cfc05d7da03b84e526e1866276f684340fa/src/ConservationLaws/burgers.jl#L20-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/09901cfc05d7da03b84e526e1866276f684340fa/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="Monday 2 October 2023 05:22">Monday 2 October 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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