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presentation.slides.html
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<!DOCTYPE html>
<html>
<head>
<meta charset="utf-8" />
<meta http-equiv="X-UA-Compatible" content="chrome=1" />
<meta name="apple-mobile-web-app-capable" content="yes" />
<meta name="apple-mobile-web-app-status-bar-style" content="black-translucent" />
<title>Presentation slides</title>
<script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.1.10/require.min.js"></script>
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/2.0.3/jquery.min.js"></script>
<!-- General and theme style sheets -->
<link rel="stylesheet" href="reveal.js/css/reveal.css">
<link rel="stylesheet" href="reveal.js/css/theme/simple.css" id="theme">
<!-- If the query includes 'print-pdf', include the PDF print sheet -->
<script>
if( window.location.search.match( /print-pdf/gi ) ) {
var link = document.createElement( 'link' );
link.rel = 'stylesheet';
link.type = 'text/css';
link.href = 'reveal.js/css/print/pdf.css';
document.getElementsByTagName( 'head' )[0].appendChild( link );
}
</script>
<!--[if lt IE 9]>
<script src="reveal.js/lib/js/html5shiv.js"></script>
<![endif]-->
<!-- Loading the mathjax macro -->
<!-- Load mathjax -->
<script src="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS_HTML"></script>
<!-- MathJax configuration -->
<script type="text/x-mathjax-config">
MathJax.Hub.Config({
tex2jax: {
inlineMath: [ ['$','$'], ["\\(","\\)"] ],
displayMath: [ ['$$','$$'], ["\\[","\\]"] ],
processEscapes: true,
processEnvironments: true
},
// Center justify equations in code and markdown cells. Elsewhere
// we use CSS to left justify single line equations in code cells.
displayAlign: 'center',
"HTML-CSS": {
styles: {'.MathJax_Display': {"margin": 0}},
linebreaks: { automatic: true }
}
});
</script>
<!-- End of mathjax configuration -->
<!-- Get Font-awesome from cdn -->
<link rel="stylesheet" href="//netdna.bootstrapcdn.com/font-awesome/4.1.0/css/font-awesome.css">
<style type="text/css">
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<h1 id="Solving-the-Shallow-Water-Equations-using-Finite-Volumes-and-Lax-Friedrichs">Solving the Shallow Water Equations using Finite Volumes and Lax-Friedrichs<a class="anchor-link" href="#Solving-the-Shallow-Water-Equations-using-Finite-Volumes-and-Lax-Friedrichs">¶</a></h1><hr>
<p>Nathan Brei</p>
<p>Technical University of Munich</p>
<p>14 December 2017</p>
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<h1 id="Shallow-Water-Equations-(1-D)">Shallow Water Equations (1-D)<a class="anchor-link" href="#Shallow-Water-Equations-(1-D)">¶</a></h1><p>Mass and momentum conservation on a fluid in a channel of unit width</p>
<ul>
<li>Assume negligible vertical velocity</li>
<li>Assume constant horizontal velocity $u(x)$ across any cross section $x$</li>
<li>Assume pressure is determined by hydrostatic pressure $p=\frac{1}{2}\rho gh^2$</li>
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<p>$\begin{bmatrix}h \\ hu \end{bmatrix}_t + \begin{bmatrix}hu \\ hu^2 + \frac{1}{2}gh^2\end{bmatrix}_x = 0 $</p>
<p>Formulate as a general conservation law in differential form: $q_t(x,t) + f(q(x,t))_x = 0$</p>
<p>$q := (h, hu) \in Q$</p>
<p>$f : (q_1, q_2) \mapsto \begin{bmatrix} q_2\\ q_2^2/q_1 + \frac{1}{2}gq_1^2\end{bmatrix}$</p>
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<h1 id="Shallow-Water-Equations-(1-D)">Shallow Water Equations (1-D)<a class="anchor-link" href="#Shallow-Water-Equations-(1-D)">¶</a></h1><p>$q := (h, hu) \in Q$</p>
<p>$f : (q_1, q_2) \mapsto \begin{bmatrix} q_2\\ q_2^2/q_1 + \frac{1}{2}gq_1^2\end{bmatrix}$</p>
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<div class=" highlight hl-julia"><pre><span class="kd">const</span> <span class="n">Float</span> <span class="o">=</span> <span class="kt">Float64</span>
<span class="kd">const</span> <span class="n">G</span> <span class="o">=</span> <span class="mf">9.81f0</span>
<span class="n">mutable</span> <span class="n">struct</span> <span class="n">Q</span>
<span class="n">h</span> <span class="p">::</span> <span class="n">Float</span>
<span class="n">hu</span> <span class="p">::</span> <span class="n">Float</span>
<span class="k">end</span>
<span class="k">function</span><span class="nf"> f</span><span class="p">(</span><span class="n">q</span> <span class="p">::</span> <span class="n">Q</span><span class="p">)</span>
<span class="n">Q</span><span class="p">(</span><span class="n">q</span><span class="o">.</span><span class="n">hu</span><span class="p">,</span> <span class="n">q</span><span class="o">.</span><span class="n">hu</span><span class="o">^</span><span class="mi">2</span><span class="o">/</span><span class="n">q</span><span class="o">.</span><span class="n">h</span> <span class="o">+</span> <span class="mf">0.5</span><span class="o">*</span><span class="n">G</span><span class="o">*</span><span class="n">q</span><span class="o">.</span><span class="n">h</span><span class="o">^</span><span class="mi">2</span><span class="p">)</span>
<span class="k">end</span>
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<pre>f (generic function with 1 method)</pre>
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<h2 id="Types-and-Methods-in-Julia">Types and Methods in Julia<a class="anchor-link" href="#Types-and-Methods-in-Julia">¶</a></h2><ul>
<li>Types are composed of primitives, structures, unions, tuples, abstract types</li>
<li>Types are parametric</li>
<li>Type annotations are optional</li>
<li>Inheritance is restricted to abstract types</li>
<li>Type system offers no encapsulation, but does support invariants and immutability</li>
<li>Methods resolved via multiple dispatch (using all arguments rather than the first)</li>
</ul>
<h2 id="Defining-a-vector-space-over-Q">Defining a vector space over Q<a class="anchor-link" href="#Defining-a-vector-space-over-Q">¶</a></h2>
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<div class=" highlight hl-julia"><pre><span class="k">import</span> <span class="n">Base</span><span class="o">.+</span>
<span class="k">import</span> <span class="n">Base</span><span class="o">.-</span>
<span class="k">import</span> <span class="n">Base</span><span class="o">.*</span>
<span class="k">function</span><span class="nf"> </span><span class="o">+(</span><span class="n">lhs</span><span class="p">::</span><span class="n">Q</span><span class="p">,</span> <span class="n">rhs</span><span class="p">::</span><span class="n">Q</span><span class="p">)</span>
<span class="n">Q</span><span class="p">(</span><span class="n">lhs</span><span class="o">.</span><span class="n">h</span> <span class="o">+</span> <span class="n">rhs</span><span class="o">.</span><span class="n">h</span><span class="p">,</span> <span class="n">lhs</span><span class="o">.</span><span class="n">hu</span> <span class="o">+</span> <span class="n">rhs</span><span class="o">.</span><span class="n">hu</span><span class="p">)</span>
<span class="k">end</span>
<span class="k">function</span><span class="nf"> </span><span class="o">-(</span><span class="n">lhs</span><span class="p">::</span><span class="n">Q</span><span class="p">,</span> <span class="n">rhs</span><span class="p">::</span><span class="n">Q</span><span class="p">)</span>
<span class="n">Q</span><span class="p">(</span><span class="n">lhs</span><span class="o">.</span><span class="n">h</span> <span class="o">-</span> <span class="n">rhs</span><span class="o">.</span><span class="n">h</span><span class="p">,</span> <span class="n">lhs</span><span class="o">.</span><span class="n">hu</span> <span class="o">-</span> <span class="n">rhs</span><span class="o">.</span><span class="n">hu</span><span class="p">)</span>
<span class="k">end</span>
<span class="k">function</span><span class="nf"> </span><span class="o">*(</span><span class="n">a</span><span class="p">::</span><span class="n">Real</span><span class="p">,</span> <span class="n">q</span><span class="p">::</span><span class="n">Q</span><span class="p">)</span>
<span class="n">Q</span><span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">q</span><span class="o">.</span><span class="n">h</span><span class="p">,</span> <span class="n">a</span><span class="o">*</span><span class="n">q</span><span class="o">.</span><span class="n">hu</span><span class="p">)</span>
<span class="k">end</span>
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<pre>* (generic function with 215 methods)</pre>
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<h2 id="CFL-Condition">CFL Condition<a class="anchor-link" href="#CFL-Condition">¶</a></h2><ul>
<li>Necessary condition for numerical stability</li>
<li>Information propagates through the spatial domain at finite speeds</li>
<li>Need {true domain of dependence} $\subset$ {numerical domain of dependence}</li>
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<p>$ \nu := \lvert\frac{\bar{u}\Delta t}{\Delta x} \rvert = \frac{\Delta t}{\Delta x} \lvert \lambda_{max} \rvert \leq 1 $</p>
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<li>Physical wave speeds determined by eigenvalues of flux Jacobian</li>
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<p>$f'(q) = \begin{bmatrix}0 & 1 \\ -u^2 + gh & 2u \end{bmatrix}$</p>
<p>$\lambda_{max} = \max\ \lvert u \pm \sqrt{gh} \rvert$</p>
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<div class=" highlight hl-julia"><pre><span class="k">function</span><span class="nf"> wavespeed</span><span class="p">(</span><span class="n">q</span> <span class="p">::</span> <span class="n">Q</span><span class="p">)</span>
<span class="n">u</span> <span class="o">=</span> <span class="n">q</span><span class="o">.</span><span class="n">hu</span> <span class="o">/</span> <span class="n">q</span><span class="o">.</span><span class="n">h</span>
<span class="n">c</span> <span class="o">=</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">G</span> <span class="o">*</span> <span class="n">q</span><span class="o">.</span><span class="n">h</span><span class="p">)</span>
<span class="n">max</span><span class="p">(</span><span class="n">abs</span><span class="p">(</span><span class="n">u</span><span class="o">-</span><span class="n">c</span><span class="p">),</span> <span class="n">abs</span><span class="p">(</span><span class="n">u</span><span class="o">+</span><span class="n">c</span><span class="p">))</span>
<span class="k">end</span>
<span class="k">function</span><span class="nf"> cfl_dt</span><span class="p">(</span><span class="n">qs</span><span class="p">::</span><span class="n">Array</span><span class="p">{</span><span class="n">Q</span><span class="p">,</span><span class="mi">1</span><span class="p">},</span> <span class="n">dx</span><span class="p">::</span><span class="n">Float</span><span class="p">,</span> <span class="n">correction</span><span class="o">=</span><span class="mf">1.0</span><span class="p">::</span><span class="n">Float</span><span class="p">)</span>
<span class="n">lambda</span> <span class="o">=</span> <span class="n">map</span><span class="p">(</span><span class="n">wavespeed</span><span class="p">,</span><span class="n">qs</span><span class="p">)</span> <span class="o">|></span> <span class="n">maximum</span>
<span class="n">dx</span> <span class="o">/</span> <span class="n">lambda</span> <span class="o">*</span> <span class="n">correction</span>
<span class="k">end</span>
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<pre>cfl_dt (generic function with 2 methods)</pre>
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<h2 id="Finite-Volumes">Finite Volumes<a class="anchor-link" href="#Finite-Volumes">¶</a></h2><p>Let $C_i := (x_{i-1/2}, x_{i+1/2})$ represent a grid cell in our spatial domain.</p>
$$\frac{d}{dt} \int_{C_i} q(x,t) dx = f(q(x_{i-1/2},t)) - f(q(x_{1+1/2},t))$$
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<p>Integrate both sides with respect to time:</p>
$$\int_{C_i} q(x,t_{n+1})dx - \int_{C_i} q(x,t_n) dx = \int_{t_n}^{t_{n+1}} f(q(x_{i-1/2},t))dt - \int_{t_n}^{t_{n+1}} f(q(x_{i+1/2},t))dt $$<p>Let $Q_{i}^{n} \approx \frac{1}{\Delta x} \int_{C_i} q(x,t_n) dx $ be the appx. average $q$ in cell $i$ at time $n$.</p>
<p>Let $F_{i+1/2}^{n} \approx \frac{1}{\Delta t} \int_{t_n}^{t_{n+1}} f(q(x_{i+1/2},t))dt$ be the appx. average flux $f$ from cell $i$ into $i+1$.</p>
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<p>Rearranging, we get a general update scheme</p>
<p>$Q_{i}^{n+1} = Q_{i}^{n} - \frac{\Delta t}{\Delta x}(F_{i+1/2}^{n} - F_{i-1/2}^{n}) $</p>
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<h1 id="Update-scheme-as-code">Update scheme as code<a class="anchor-link" href="#Update-scheme-as-code">¶</a></h1><p>$Q_{i}^{n+1} = Q_{i}^{n} - \frac{\Delta t}{\Delta x}(F_{i+1/2}^{n} - F_{i-1/2}^{n}) $</p>
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<div class=" highlight hl-julia"><pre><span class="k">function</span><span class="nf"> update_qs</span><span class="o">!</span><span class="p">(</span><span class="n">qs</span><span class="p">::</span><span class="n">Array</span><span class="p">{</span><span class="n">Q</span><span class="p">,</span><span class="mi">1</span><span class="p">},</span> <span class="n">Fl</span><span class="p">::</span><span class="n">Array</span><span class="p">{</span><span class="n">Q</span><span class="p">,</span><span class="mi">1</span><span class="p">},</span> <span class="n">Fr</span><span class="p">::</span><span class="n">Array</span><span class="p">{</span><span class="n">Q</span><span class="p">,</span><span class="mi">1</span><span class="p">},</span>
<span class="n">ncells</span><span class="p">::</span><span class="kt">Int</span><span class="p">,</span> <span class="n">dx</span><span class="p">::</span><span class="n">Float</span><span class="p">,</span> <span class="n">dt</span><span class="p">::</span><span class="n">Float</span><span class="p">)</span>
<span class="k">for</span> <span class="n">x</span> <span class="o">=</span> <span class="mi">2</span><span class="p">:</span><span class="n">ncells</span><span class="o">+</span><span class="mi">1</span>
<span class="n">qs</span><span class="p">[</span><span class="n">x</span><span class="p">]</span> <span class="o">-=</span> <span class="n">dt</span><span class="o">/</span><span class="n">dx</span> <span class="o">*</span> <span class="p">(</span><span class="n">Fr</span><span class="p">[</span><span class="n">x</span><span class="p">]</span> <span class="o">+</span> <span class="n">Fl</span><span class="p">[</span><span class="n">x</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span>
<span class="k">end</span>
<span class="k">end</span>
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<pre>update_qs! (generic function with 1 method)</pre>
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<table>
<thead><tr>
<th>Math</th>
<th>Code</th>
<th>Description</th>
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<tr>
<td>$Q_{i}^{n}$</td>
<td><code>qs[x]</code></td>
<td>1D array of Q containing solution states</td>
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<tr>
<td>$F_{i+1/2}^{n}$</td>
<td><code>Fr[x]</code></td>
<td>1D array of Q containing flux flowing right </td>
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<tr>
<td>$F_{i-1/2}^{n}$</td>
<td><code>Fl[x]</code></td>
<td>1D array of Q containing flux flowing left</td>
</tr>
</tbody>
</table>
<ul>
<li>Each buffer is of length <code>ncells+2</code>, accounting for ghost cells at either end</li>
<li>Functions suffixed with a ! update one or more of their buffers in-place</li>
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<h1 id="Timestepping-loop">Timestepping loop<a class="anchor-link" href="#Timestepping-loop">¶</a></h1>
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<div class=" highlight hl-julia"><pre><span class="k">function</span><span class="nf"> run_experiment</span><span class="p">(</span><span class="n">stoptime</span><span class="p">,</span> <span class="n">ncells</span><span class="p">::</span><span class="kt">Int</span><span class="p">,</span> <span class="n">choose_dt</span><span class="p">,</span>
<span class="n">compute_fluxes</span><span class="o">!</span><span class="p">,</span> <span class="n">apply_bcs</span><span class="o">!</span><span class="p">,</span> <span class="n">apply_ics</span><span class="p">)</span>
<span class="n">qs</span> <span class="o">=</span> <span class="n">apply_ics</span><span class="p">(</span><span class="n">ncells</span><span class="p">)</span>
<span class="n">Fl</span> <span class="o">=</span> <span class="n">Array</span><span class="p">{</span><span class="n">Q</span><span class="p">,</span><span class="mi">1</span><span class="p">}(</span><span class="n">ncells</span><span class="o">+</span><span class="mi">2</span><span class="p">)</span>
<span class="n">Fr</span> <span class="o">=</span> <span class="n">Array</span><span class="p">{</span><span class="n">Q</span><span class="p">,</span><span class="mi">1</span><span class="p">}(</span><span class="n">ncells</span><span class="o">+</span><span class="mi">2</span><span class="p">)</span>
<span class="n">currenttime</span> <span class="o">=</span> <span class="mi">0</span>
<span class="n">timesteps</span> <span class="o">=</span> <span class="mi">0</span>
<span class="n">dx</span> <span class="o">=</span> <span class="n">Float</span><span class="p">(</span><span class="mf">1000.0</span><span class="o">/</span><span class="n">ncells</span><span class="p">)</span>
<span class="k">while</span> <span class="n">currenttime</span> <span class="o"><</span> <span class="n">stoptime</span>
<span class="n">apply_bcs</span><span class="o">!</span><span class="p">(</span><span class="n">qs</span><span class="p">)</span>
<span class="n">dt</span> <span class="o">=</span> <span class="n">choose_dt</span><span class="p">(</span><span class="n">qs</span><span class="p">,</span> <span class="n">dx</span><span class="p">)</span>
<span class="n">compute_fluxes</span><span class="o">!</span><span class="p">(</span><span class="n">qs</span><span class="p">,</span> <span class="n">Fl</span><span class="p">,</span> <span class="n">Fr</span><span class="p">,</span> <span class="n">ncells</span><span class="p">,</span> <span class="n">dx</span><span class="p">,</span> <span class="n">dt</span><span class="p">)</span>
<span class="n">update_qs</span><span class="o">!</span><span class="p">(</span><span class="n">qs</span><span class="p">,</span> <span class="n">Fl</span><span class="p">,</span> <span class="n">Fr</span><span class="p">,</span> <span class="n">ncells</span><span class="p">,</span> <span class="n">dx</span><span class="p">,</span> <span class="n">dt</span><span class="p">)</span>
<span class="n">currenttime</span> <span class="o">+=</span> <span class="n">dt</span>
<span class="n">timesteps</span> <span class="o">+=</span> <span class="mi">1</span>
<span class="k">end</span>
<span class="k">return</span> <span class="p">(</span><span class="n">qs</span><span class="p">,</span> <span class="n">currenttime</span><span class="p">,</span> <span class="n">timesteps</span><span class="p">)</span>
<span class="k">end</span>
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<pre>run_experiment (generic function with 1 method)</pre>
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<h1 id="Timestepping-loop-for-animations">Timestepping loop for animations<a class="anchor-link" href="#Timestepping-loop-for-animations">¶</a></h1><ul>
<li>Outer timesteps correspond to clean increments of physical time</li>
<li>An outer loop controls outer timesteps by iterating over stoptime</li>
<li>Inner timesteps correspond to increments of dt</li>
<li>Inner timesteps go up to stop time exactly</li>
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<div class=" highlight hl-julia"><pre><span class="k">function</span><span class="nf"> run_experiment</span><span class="o">!</span><span class="p">(</span><span class="n">qs</span><span class="p">,</span> <span class="n">Fl</span><span class="p">,</span> <span class="n">Fr</span><span class="p">,</span>
<span class="n">currenttime</span><span class="p">::</span><span class="n">Real</span><span class="p">,</span> <span class="n">stoptime</span><span class="p">::</span><span class="n">Real</span><span class="p">,</span>
<span class="n">ncells</span><span class="p">::</span><span class="kt">Int</span><span class="p">,</span>
<span class="n">choose_dt</span><span class="p">,</span> <span class="n">compute_fluxes</span><span class="o">!</span><span class="p">,</span> <span class="n">apply_bcs</span><span class="o">!</span><span class="p">)</span>
<span class="n">timesteps</span> <span class="o">=</span> <span class="mi">0</span>
<span class="n">dx</span> <span class="o">=</span> <span class="n">Float</span><span class="p">(</span><span class="mf">1000.0</span><span class="o">/</span><span class="n">ncells</span><span class="p">)</span>
<span class="k">while</span> <span class="n">currenttime</span> <span class="o"><</span> <span class="n">stoptime</span>
<span class="n">apply_bcs</span><span class="o">!</span><span class="p">(</span><span class="n">qs</span><span class="p">)</span>
<span class="n">dt</span> <span class="o">=</span> <span class="n">choose_dt</span><span class="p">(</span><span class="n">qs</span><span class="p">,</span> <span class="n">dx</span><span class="p">)</span>
<span class="k">if</span> <span class="n">currenttime</span> <span class="o">+</span> <span class="n">dt</span> <span class="o">></span> <span class="n">stoptime</span>
<span class="n">dt</span> <span class="o">=</span> <span class="n">stoptime</span> <span class="o">-</span> <span class="n">currenttime</span>
<span class="k">end</span>
<span class="n">compute_fluxes</span><span class="o">!</span><span class="p">(</span><span class="n">qs</span><span class="p">,</span> <span class="n">Fl</span><span class="p">,</span> <span class="n">Fr</span><span class="p">,</span> <span class="n">ncells</span><span class="p">,</span> <span class="n">dx</span><span class="p">,</span> <span class="n">dt</span><span class="p">)</span>
<span class="n">update_qs</span><span class="o">!</span><span class="p">(</span><span class="n">qs</span><span class="p">,</span> <span class="n">Fl</span><span class="p">,</span> <span class="n">Fr</span><span class="p">,</span> <span class="n">ncells</span><span class="p">,</span> <span class="n">dx</span><span class="p">,</span> <span class="n">dt</span><span class="p">)</span>
<span class="n">currenttime</span> <span class="o">+=</span> <span class="n">dt</span>
<span class="n">timesteps</span> <span class="o">+=</span> <span class="mi">1</span>
<span class="k">end</span>
<span class="k">return</span> <span class="p">(</span><span class="n">qs</span><span class="p">,</span> <span class="n">currenttime</span><span class="p">,</span> <span class="n">timesteps</span><span class="p">)</span>
<span class="k">end</span>
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<pre>run_experiment! (generic function with 2 methods)</pre>
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<h1 id="Lax-Friedrichs-by-accident">Lax-Friedrichs by accident<a class="anchor-link" href="#Lax-Friedrichs-by-accident">¶</a></h1><p>Consider the simplest choice of F:</p>
<p>$ F_{i-1/2} = \frac{1}{2}[f(Q_{i-1} + f(Q_i)] $</p>
<p>$ Q_i^{n+1} = Q_i^n - \frac{\Delta t}{2 \Delta x} [f(Q_{i+1}^n) - f(Q_{i-1}^n)]$</p>
<p>Unfortunately this is unstable.</p>
<p>Instead, replace $Q_i^n$ with $\frac{1}{2}(Q_{i-1}^n + Q_{i+1}^n)$</p>
<p>$ Q_i^{n+1} = \frac{1}{2}(Q_{i-1}^n + Q_{i+1}^n) - \frac{\Delta t}{2 \Delta x} [f(Q_{i+1}^n) - f(Q_{i-1}^n)]$</p>
<p>Rearrange to get a flux update function:</p>
<p>$ F_{i-1/2} = \frac{1}{2} [ f(Q_{i-1}) + f(Q_i) - a(Q_i - Q_{i-1})]$</p>
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<h1 id="Lax-Friedrichs-Solver">Lax-Friedrichs Solver<a class="anchor-link" href="#Lax-Friedrichs-Solver">¶</a></h1><ul>
<li>Input: Buffers for $q(x), F_l(x), F_r(x)$, cell and timestep sizes</li>
<li>Output: Updated inter-cell flux buffers</li>
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<div class=" highlight hl-julia"><pre><span class="k">function</span><span class="nf"> lxf</span><span class="o">!</span><span class="p">(</span><span class="n">qs</span><span class="p">::</span><span class="n">Array</span><span class="p">{</span><span class="n">Q</span><span class="p">,</span><span class="mi">1</span><span class="p">},</span> <span class="n">Fl</span><span class="p">::</span><span class="n">Array</span><span class="p">{</span><span class="n">Q</span><span class="p">,</span><span class="mi">1</span><span class="p">},</span> <span class="n">Fr</span><span class="p">::</span><span class="n">Array</span><span class="p">{</span><span class="n">Q</span><span class="p">,</span><span class="mi">1</span><span class="p">},</span>
<span class="n">ncells</span><span class="p">::</span><span class="kt">Int</span><span class="p">,</span> <span class="n">dx</span><span class="p">::</span><span class="n">Float</span><span class="p">,</span> <span class="n">dt</span><span class="p">::</span><span class="n">Float</span><span class="p">)</span>
<span class="n">a</span> <span class="o">=</span> <span class="n">dx</span><span class="o">/</span><span class="n">dt</span>
<span class="k">for</span> <span class="n">x</span> <span class="o">=</span> <span class="mi">2</span><span class="p">:</span><span class="n">ncells</span><span class="o">+</span><span class="mi">2</span>
<span class="n">ql</span> <span class="o">=</span> <span class="n">qs</span><span class="p">[</span><span class="n">x</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
<span class="n">qr</span> <span class="o">=</span> <span class="n">qs</span><span class="p">[</span><span class="n">x</span><span class="p">]</span>
<span class="n">fl</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">ql</span><span class="p">)</span>
<span class="n">fr</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">qr</span><span class="p">)</span>
<span class="n">Fr</span><span class="p">[</span><span class="n">x</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="mf">0.5</span><span class="o">*</span><span class="p">((</span><span class="n">fr</span> <span class="o">-</span> <span class="n">fl</span><span class="p">)</span> <span class="o">-</span> <span class="n">a</span><span class="o">*</span><span class="p">(</span><span class="n">qr</span> <span class="o">-</span> <span class="n">ql</span><span class="p">))</span>
<span class="n">Fl</span><span class="p">[</span><span class="n">x</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="mf">0.5</span><span class="o">*</span><span class="p">((</span><span class="n">fr</span> <span class="o">-</span> <span class="n">fl</span><span class="p">)</span> <span class="o">+</span> <span class="n">a</span><span class="o">*</span><span class="p">(</span><span class="n">qr</span> <span class="o">-</span> <span class="n">ql</span><span class="p">))</span>
<span class="k">end</span>
<span class="k">end</span>
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<pre>lxf! (generic function with 1 method)</pre>
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<h1 id="Gaussian-Initial-Condition">Gaussian Initial Condition<a class="anchor-link" href="#Gaussian-Initial-Condition">¶</a></h1>
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<div class=" highlight hl-julia"><pre><span class="k">function</span><span class="nf"> gaussian_ics</span><span class="p">(</span><span class="n">ncells</span><span class="p">::</span><span class="kt">Int</span><span class="p">)</span>
<span class="n">qs</span> <span class="o">=</span> <span class="n">Array</span><span class="p">{</span><span class="n">Q</span><span class="p">,</span><span class="mi">1</span><span class="p">}(</span><span class="n">ncells</span><span class="o">+</span><span class="mi">2</span><span class="p">)</span>
<span class="n">mid</span> <span class="o">=</span> <span class="n">ncells</span> <span class="o">/</span> <span class="mf">2.0</span> <span class="o">+</span> <span class="mi">1</span>
<span class="k">for</span> <span class="n">x</span> <span class="o">=</span> <span class="mi">1</span><span class="p">:</span><span class="n">ncells</span><span class="o">+</span><span class="mi">2</span>
<span class="n">pos_rel</span> <span class="o">=</span> <span class="p">(</span><span class="n">mid</span><span class="o">-</span><span class="n">x</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="n">ncells</span><span class="o">/</span><span class="mf">10.0</span><span class="p">)</span>
<span class="n">qs</span><span class="p">[</span><span class="n">x</span><span class="p">]</span> <span class="o">=</span> <span class="n">Q</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">pos_rel</span><span class="o">^</span><span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">)</span>
<span class="k">end</span>
<span class="k">return</span> <span class="n">qs</span>
<span class="k">end</span>
<span class="n">plotxh</span><span class="p">(</span><span class="n">gaussian_ics</span><span class="p">(</span><span class="mi">100</span><span class="p">),</span> <span class="s">"Gaussian initial conditions"</span><span class="p">)</span>
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" 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<h1 id="Breaking-Dam-Initial-Conditions">Breaking Dam Initial Conditions<a class="anchor-link" href="#Breaking-Dam-Initial-Conditions">¶</a></h1>
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<div class="prompt input_prompt">In [107]:</div>
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<div class=" highlight hl-julia"><pre><span class="k">function</span><span class="nf"> breakingdam_ics</span><span class="p">(</span><span class="n">ncells</span><span class="p">::</span><span class="kt">Int</span><span class="p">)</span>
<span class="p">[</span><span class="n">x</span> <span class="o"><</span> <span class="n">ncells</span><span class="o">/</span><span class="mi">2</span><span class="o">+</span><span class="mi">1</span> <span class="o">?</span> <span class="n">Q</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">0</span><span class="p">)</span> <span class="p">:</span> <span class="n">Q</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="k">in</span> <span class="mi">1</span><span class="p">:</span><span class="n">ncells</span><span class="o">+</span><span class="mi">2</span><span class="p">]</span>
<span class="k">end</span>
<span class="n">plotxh</span><span class="p">(</span><span class="n">breakingdam_ics</span><span class="p">(</span><span class="mi">100</span><span class="p">),</span> <span class="s">"Breaking dam initial conditions"</span><span class="p">)</span>
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<div class="output_area"><div class="prompt output_prompt">Out[107]:</div>
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" 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<h1 id="Boundary-Conditions">Boundary Conditions<a class="anchor-link" href="#Boundary-Conditions">¶</a></h1>
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<div class="prompt input_prompt">In [176]:</div>
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<div class=" highlight hl-julia"><pre><span class="k">function</span><span class="nf"> outflow_bcs</span><span class="o">!</span><span class="p">(</span><span class="n">qs</span><span class="p">::</span><span class="n">Array</span><span class="p">{</span><span class="n">Q</span><span class="p">,</span><span class="mi">1</span><span class="p">})</span>
<span class="n">qs</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">qs</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span>
<span class="n">qs</span><span class="p">[</span><span class="k">end</span><span class="p">]</span> <span class="o">=</span> <span class="n">qs</span><span class="p">[</span><span class="k">end</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
<span class="k">end</span>
<span class="k">function</span><span class="nf"> periodic_bcs</span><span class="o">!</span><span class="p">(</span><span class="n">qs</span><span class="p">::</span><span class="n">Array</span><span class="p">{</span><span class="n">Q</span><span class="p">,</span><span class="mi">1</span><span class="p">})</span>
<span class="n">qs</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">qs</span><span class="p">[</span><span class="k">end</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
<span class="n">qs</span><span class="p">[</span><span class="k">end</span><span class="p">]</span> <span class="o">=</span> <span class="n">qs</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span>
<span class="k">end</span>
<span class="k">function</span><span class="nf"> reflecting_bcs</span><span class="o">!</span><span class="p">(</span><span class="n">qs</span><span class="p">::</span><span class="n">Array</span><span class="p">{</span><span class="n">Q</span><span class="p">,</span><span class="mi">1</span><span class="p">})</span>
<span class="n">qs</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">h</span> <span class="o">=</span> <span class="n">qs</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">h</span>
<span class="n">qs</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">hu</span> <span class="o">=</span> <span class="o">-</span><span class="n">qs</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">hu</span>
<span class="n">qs</span><span class="p">[</span><span class="k">end</span><span class="p">]</span><span class="o">.</span><span class="n">h</span> <span class="o">=</span> <span class="n">qs</span><span class="p">[</span><span class="k">end</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">h</span>
<span class="n">qs</span><span class="p">[</span><span class="k">end</span><span class="p">]</span><span class="o">.</span><span class="n">hu</span> <span class="o">=</span> <span class="o">-</span><span class="n">qs</span><span class="p">[</span><span class="k">end</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">hu</span>
<span class="k">end</span>
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<div class="output_area"><div class="prompt output_prompt">Out[176]:</div>
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<pre>reflecting_bcs! (generic function with 1 method)</pre>
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<h1 id="Timestep-strategies">Timestep strategies<a class="anchor-link" href="#Timestep-strategies">¶</a></h1><ul>
<li>Idea: Strategies are simply functions</li>
<li>Timestep strategy always has the signature $ ([q], \Delta x) \mapsto \Delta t $</li>
<li>Use <em>currying</em> to pass configuration information</li>
<li>Use <em>closures</em> to hold any state that may change at runtime </li>
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<div class=" highlight hl-julia"><pre><span class="k">function</span><span class="nf"> make_const_dt</span><span class="p">(</span><span class="n">dt</span><span class="p">)</span>
<span class="k">return</span> <span class="p">(</span><span class="n">qs</span><span class="p">,</span> <span class="n">dx</span><span class="p">)</span> <span class="o">-></span> <span class="n">dt</span>
<span class="k">end</span>
<span class="k">function</span><span class="nf"> make_cfl_dt</span><span class="p">(</span><span class="n">correction</span><span class="p">::</span><span class="n">Float</span><span class="p">)</span>
<span class="k">return</span> <span class="p">(</span><span class="n">qs</span><span class="p">,</span> <span class="n">dx</span><span class="p">)</span> <span class="o">-></span> <span class="n">cfl_dt</span><span class="p">(</span><span class="n">qs</span><span class="p">,</span> <span class="n">dx</span><span class="p">,</span> <span class="n">correction</span><span class="p">)</span>
<span class="k">end</span>
<span class="k">function</span><span class="nf"> make_lagging_dt</span><span class="p">(</span><span class="n">dt0</span><span class="p">,</span> <span class="n">correction</span><span class="p">)</span>
<span class="n">dt</span> <span class="o">=</span> <span class="n">dt0</span>
<span class="k">function</span><span class="nf"> lagging_dt</span><span class="p">(</span><span class="n">qs</span><span class="p">,</span> <span class="n">dx</span><span class="p">)</span>
<span class="n">result</span> <span class="o">=</span> <span class="n">dt</span>
<span class="n">dt</span> <span class="o">=</span> <span class="n">cfl_dt</span><span class="p">(</span><span class="n">qs</span><span class="p">,</span> <span class="n">dx</span><span class="p">,</span> <span class="n">correction</span><span class="p">)</span>
<span class="k">return</span> <span class="n">result</span>
<span class="k">end</span>
<span class="k">return</span> <span class="n">lagging_dt</span>
<span class="k">end</span>
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<pre>make_lagging_dt (generic function with 1 method)</pre>
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<h1 id="Lax-Friedrichs-solutions">Lax-Friedrichs solutions<a class="anchor-link" href="#Lax-Friedrichs-solutions">¶</a></h1>
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<div class=" highlight hl-julia"><pre><span class="n">figure23</span>
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