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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>Advection · EarthSciMLBase.jl</title><meta name="title" content="Advection · EarthSciMLBase.jl"/><meta property="og:title" content="Advection · EarthSciMLBase.jl"/><meta property="twitter:title" content="Advection · EarthSciMLBase.jl"/><meta name="description" content="Documentation for EarthSciMLBase.jl."/><meta property="og:description" content="Documentation for EarthSciMLBase.jl."/><meta property="twitter:description" content="Documentation for EarthSciMLBase.jl."/><meta property="og:url" content="https://EarthSciML.github.io/EarthSciMLBase.jl/advection/"/><meta property="twitter:url" content="https://EarthSciML.github.io/EarthSciMLBase.jl/advection/"/><link rel="canonical" href="https://EarthSciML.github.io/EarthSciMLBase.jl/advection/"/><script data-outdated-warner src="../assets/warner.js"></script><link 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src="../assets/themeswap.js"></script></head><body><div id="documenter"><nav class="docs-sidebar"><div class="docs-package-name"><span class="docs-autofit"><a href="../">EarthSciMLBase.jl</a></span></div><button class="docs-search-query input is-rounded is-small is-clickable my-2 mx-auto py-1 px-2" id="documenter-search-query">Search docs (Ctrl + /)</button><ul class="docs-menu"><li><a class="tocitem" href="../">Home</a></li><li><a class="tocitem" href="../composition/">Composition</a></li><li><a class="tocitem" href="../operator_compose/">Operator Composition</a></li><li><a class="tocitem" href="../param_to_var/">Parameter Replacement</a></li><li><a class="tocitem" href="../icbc/">Initial and Boundary Conditions</a></li><li class="is-active"><a class="tocitem" href>Advection</a></li><li><a class="tocitem" href="../coord_transforms/">Coordinate Transforms</a></li><li><a class="tocitem" href="../operator/">Operators</a></li><li><span class="tocitem">Examples</span><ul><li><a class="tocitem" href="../example_all_together/">All Together</a></li></ul></li><li><a class="tocitem" href="../api/">API Reference</a></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"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li class="is-active"><a href>Advection</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Advection</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/EarthSciML/EarthSciMLBase.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/EarthSciML/EarthSciMLBase.jl/blob/main/docs/src/advection.md#" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Advection"><a class="docs-heading-anchor" href="#Advection">Advection</a><a id="Advection-1"></a><a class="docs-heading-anchor-permalink" href="#Advection" title="Permalink"></a></h1><p>The <code>Advection</code> function adds advection to a system of equations. This is useful for modeling the transport of a substance by a fluid. Advection is implemented with the <a href="#Advection"><code>Advection</code></a> type.</p><p>!warning     Fully symbolic partial differential equations like those shown here don&#39;t currently work on domains that have a large number of grid cells. See <a href="https://docs.sciml.ai/MethodOfLines/stable/performance/">here</a> for additional information.</p><p>To demonstrate how this can work, we will start with a simple system of equations:</p><pre><code class="language-julia hljs">using EarthSciMLBase, ModelingToolkit
+using ModelingToolkit: t_nounits, D_nounits
+t = t_nounits
+D = D_nounits
+
+function ExampleSys()
+    @variables y(t)
+    @parameters p=2.0
+    ODESystem([D(y) ~ p], t; name=:ExampleSys)
+end
+
+ExampleSys()</code></pre><p class="math-container">\[ \begin{align}
+\frac{\mathrm{d} y\left( t \right)}{\mathrm{d}t} &amp;= p
+\end{align}
+ \]</p><p>We also need to create our initial and boundary conditions.</p><pre><code class="language-julia hljs">using DomainSets
+@parameters x
+domain = DomainInfo(constIC(0.0, t ∈ Interval(0, 1.0)), constBC(1.0, x ∈ Interval(0, 1.0)))</code></pre><p>Now we convert add advection to each of the state variables. We&#39;re also adding a constant wind (<a href="../api/#EarthSciMLBase.ConstantWind-Tuple{Any, Vararg{Any}}"><code>ConstantWind</code></a>) in the x-direction, with a speed of 1.0.</p><pre><code class="language-julia hljs">sys_advection = couple(ExampleSys(), domain, ConstantWind(t, 1.0), Advection())
+sys_mtk = convert(PDESystem, sys_advection)</code></pre><p class="math-container">\[ \begin{align}
+\frac{\mathrm{d}}{\mathrm{d}t} \mathtt{ExampleSys.y}\left( t, x \right) &amp;= \mathtt{ExampleSys.p} - \mathtt{MeanWind.v\_x}\left( t, x \right) \frac{\mathrm{d}}{\mathrm{d}x} \mathtt{ExampleSys.y}\left( t, x \right) \\
+\mathtt{MeanWind.v\_x}\left( t, x \right) &amp;= \mathtt{ConstantWind.v\_1}\left( t, x \right) \\
+\mathtt{ConstantWind.v\_1}\left( t, x \right) &amp;= \mathtt{ConstantWind.c\_v1}
+\end{align}
+ \]</p><p>Finally, we can discretize the system and solve it:</p><pre><code class="language-julia hljs">using MethodOfLines, DifferentialEquations, Plots
+discretization = MOLFiniteDifference([x=&gt;10], t, approx_order=2)
+@time prob = discretize(sys_mtk, discretization)
+@time sol = solve(prob, Tsit5(), saveat=0.1)
+
+
+# Plot the solution.
+discrete_x = sol[x]
+discrete_t = sol[t]
+soly = sol[sys_mtk.dvs[3]]
+anim = @animate for k in 1:length(discrete_t)
+    plot(discrete_x, soly[k, 1:end], title=&quot;t=\$(discrete_t[k])&quot;, ylim=(0,2.5), lab=:none)
+end
+gif(anim, fps = 8)</code></pre><img src="09a42f74.gif" alt="Example block output"/></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../icbc/">« Initial and Boundary Conditions</a><a class="docs-footer-nextpage" href="../coord_transforms/">Coordinate Transforms »</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="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.7.0 on <span class="colophon-date" title="Thursday 31 October 2024 23:16">Thursday 31 October 2024</span>. Using Julia version 1.11.1.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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