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<h1 class="title d-none d-lg-block"><span class="chapter-number">50</span>&nbsp; <span class="chapter-title">The problem-algorithm-solve interface</span></h1>
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<p>This section uses these add-on packages:</p>
<div class="sourceCode cell-code" id="cb1"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb1-1"><a href="#cb1-1" aria-hidden="true" tabindex="-1"></a><span class="im">using</span> <span class="bu">Plots</span></span>
<span id="cb1-2"><a href="#cb1-2" aria-hidden="true" tabindex="-1"></a><span class="im">using</span> <span class="bu">MonteCarloMeasurements</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<hr>
<p>The <a href="https://github.com/SciML">DifferentialEquations.jl</a> package is an entry point to a suite of <code>Julia</code> packages for numerically solving differential equations in <code>Julia</code> and other languages. A common interface is implemented that flexibly adjusts to the many different problems and algorithms covered by this suite of packages. In this section, we review a very informative <a href="https://discourse.julialang.org/t/function-depending-on-the-global-variable-inside-module/64322/10">post</a> by discourse user <code>@genkuroki</code> which very nicely demonstrates the usefulness of the problem-algorithm-solve approach used with <code>DifferentialEquations.jl</code>. We slightly modify the presentation below for our needs, but suggest a perusal of the original post.</p>
<section id="example-freefall" class="level5">
<h5 class="anchored" data-anchor-id="example-freefall">Example: FreeFall</h5>
<p>The motion of an object under a uniform gravitational field is of interest.</p>
<p>The parameters that govern the equation of motions are the gravitational constant, <code>g</code>; the initial height, <code>y0</code>; and the initial velocity, <code>v0</code>. The time span for which a solution is sought is <code>tspan</code>.</p>
<p>A problem consists of these parameters. Typical <code>Julia</code> usage would be to create a structure to hold the parameters, which may be done as follows:</p>
<div class="cell" data-execution_count="4">
<div class="sourceCode cell-code" id="cb2"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb2-1"><a href="#cb2-1" aria-hidden="true" tabindex="-1"></a><span class="kw">struct</span> Problem{G, Y0, V0, TS}</span>
<span id="cb2-2"><a href="#cb2-2" aria-hidden="true" tabindex="-1"></a> g<span class="op">::</span><span class="dt">G</span></span>
<span id="cb2-3"><a href="#cb2-3" aria-hidden="true" tabindex="-1"></a> y0<span class="op">::</span><span class="dt">Y0</span></span>
<span id="cb2-4"><a href="#cb2-4" aria-hidden="true" tabindex="-1"></a> v0<span class="op">::</span><span class="dt">V0</span></span>
<span id="cb2-5"><a href="#cb2-5" aria-hidden="true" tabindex="-1"></a> tspan<span class="op">::</span><span class="dt">TS</span></span>
<span id="cb2-6"><a href="#cb2-6" aria-hidden="true" tabindex="-1"></a><span class="kw">end</span></span>
<span id="cb2-7"><a href="#cb2-7" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb2-8"><a href="#cb2-8" aria-hidden="true" tabindex="-1"></a><span class="fu">Problem</span>(;g<span class="op">=</span><span class="fl">9.80665</span>, y0<span class="op">=</span><span class="fl">0.0</span>, v0<span class="op">=</span><span class="fl">30.0</span>, tspan<span class="op">=</span>(<span class="fl">0.0</span>,<span class="fl">8.0</span>)) <span class="op">=</span> <span class="fu">Problem</span>(g, y0, v0, tspan)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-display" data-execution_count="5">
<pre><code>Problem</code></pre>
</div>
</div>
<p>The above creates a type, <code>Problem</code>, <em>and</em> a default constructor with default values. (The original uses a more sophisticated setup that allows the two things above to be combined.)</p>
<p>Just calling <code>Problem()</code> will create a problem suitable for the earth, passing different values for <code>g</code> would be possible for other planets.</p>
<p>To solve differential equations there are many different possible algorithms. Here is the construction of two types to indicate two algorithms:</p>
<div class="cell" data-execution_count="5">
<div class="sourceCode cell-code" id="cb4"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb4-1"><a href="#cb4-1" aria-hidden="true" tabindex="-1"></a><span class="kw">struct</span> EulerMethod{T}</span>
<span id="cb4-2"><a href="#cb4-2" aria-hidden="true" tabindex="-1"></a> dt<span class="op">::</span><span class="dt">T</span></span>
<span id="cb4-3"><a href="#cb4-3" aria-hidden="true" tabindex="-1"></a><span class="kw">end</span></span>
<span id="cb4-4"><a href="#cb4-4" aria-hidden="true" tabindex="-1"></a><span class="fu">EulerMethod</span>(; dt<span class="op">=</span><span class="fl">0.1</span>) <span class="op">=</span> <span class="fu">EulerMethod</span>(dt)</span>
<span id="cb4-5"><a href="#cb4-5" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb4-6"><a href="#cb4-6" aria-hidden="true" tabindex="-1"></a><span class="kw">struct</span> ExactFormula{T}</span>
<span id="cb4-7"><a href="#cb4-7" aria-hidden="true" tabindex="-1"></a> dt<span class="op">::</span><span class="dt">T</span></span>
<span id="cb4-8"><a href="#cb4-8" aria-hidden="true" tabindex="-1"></a><span class="kw">end</span></span>
<span id="cb4-9"><a href="#cb4-9" aria-hidden="true" tabindex="-1"></a><span class="fu">ExactFormula</span>(; dt<span class="op">=</span><span class="fl">0.1</span>) <span class="op">=</span> <span class="fu">ExactFormula</span>(dt)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-display" data-execution_count="6">
<pre><code>ExactFormula</code></pre>
</div>
</div>
<p>The above just specifies a type for dispatch –- the directions indicating what code to use to solve the problem. As seen, each specifies a size for a time step with default of <code>0.1</code>.</p>
<p>A type for solutions is useful for different <code>show</code> methods or other methods. One can be created through:</p>
<div class="sourceCode cell-code" id="cb6"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb6-1"><a href="#cb6-1" aria-hidden="true" tabindex="-1"></a><span class="kw">struct</span> Solution{Y, V, T, P<span class="op">&lt;:</span><span class="dt">Problem</span>, A}</span>
<span id="cb6-2"><a href="#cb6-2" aria-hidden="true" tabindex="-1"></a> y<span class="op">::</span><span class="dt">Y</span></span>
<span id="cb6-3"><a href="#cb6-3" aria-hidden="true" tabindex="-1"></a> v<span class="op">::</span><span class="dt">V</span></span>
<span id="cb6-4"><a href="#cb6-4" aria-hidden="true" tabindex="-1"></a> t<span class="op">::</span><span class="dt">T</span></span>
<span id="cb6-5"><a href="#cb6-5" aria-hidden="true" tabindex="-1"></a> prob<span class="op">::</span><span class="dt">P</span></span>
<span id="cb6-6"><a href="#cb6-6" aria-hidden="true" tabindex="-1"></a> alg<span class="op">::</span><span class="dt">A</span></span>
<span id="cb6-7"><a href="#cb6-7" aria-hidden="true" tabindex="-1"></a><span class="kw">end</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<p>The different algorithms then can be implemented as part of a generic <code>solve</code> function. Following the post we have:</p>
<div class="cell" data-execution_count="7">
<div class="sourceCode cell-code" id="cb7"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb7-1"><a href="#cb7-1" aria-hidden="true" tabindex="-1"></a><span class="fu">solve</span>(prob<span class="op">::</span><span class="dt">Problem</span>) <span class="op">=</span> <span class="fu">solve</span>(prob, <span class="fu">default_algorithm</span>(prob))</span>
<span id="cb7-2"><a href="#cb7-2" aria-hidden="true" tabindex="-1"></a><span class="fu">default_algorithm</span>(prob<span class="op">::</span><span class="dt">Problem</span>) <span class="op">=</span> <span class="fu">EulerMethod</span>()</span>
<span id="cb7-3"><a href="#cb7-3" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb7-4"><a href="#cb7-4" aria-hidden="true" tabindex="-1"></a><span class="kw">function</span> <span class="fu">solve</span>(prob<span class="op">::</span><span class="dt">Problem</span>, alg<span class="op">::</span><span class="dt">ExactFormula</span>)</span>
<span id="cb7-5"><a href="#cb7-5" aria-hidden="true" tabindex="-1"></a> g, y0, v0, tspan <span class="op">=</span> prob.g, prob.y0, prob.v0, prob.tspan</span>
<span id="cb7-6"><a href="#cb7-6" aria-hidden="true" tabindex="-1"></a> dt <span class="op">=</span> alg.dt</span>
<span id="cb7-7"><a href="#cb7-7" aria-hidden="true" tabindex="-1"></a> t0, t1 <span class="op">=</span> tspan</span>
<span id="cb7-8"><a href="#cb7-8" aria-hidden="true" tabindex="-1"></a> t <span class="op">=</span> <span class="fu">range</span>(t0, t1 <span class="op">+</span> dt<span class="op">/</span><span class="fl">2</span>; step <span class="op">=</span> dt)</span>
<span id="cb7-9"><a href="#cb7-9" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb7-10"><a href="#cb7-10" aria-hidden="true" tabindex="-1"></a> <span class="fu">y</span>(t) <span class="op">=</span> y0 <span class="op">+</span> <span class="fu">v0*</span>(t <span class="op">-</span> t0) <span class="op">-</span> <span class="fu">g*</span>(t <span class="op">-</span> t0)<span class="op">^</span><span class="fl">2</span><span class="op">/</span><span class="fl">2</span></span>
<span id="cb7-11"><a href="#cb7-11" aria-hidden="true" tabindex="-1"></a> <span class="fu">v</span>(t) <span class="op">=</span> v0 <span class="op">-</span> <span class="fu">g*</span>(t <span class="op">-</span> t0)</span>
<span id="cb7-12"><a href="#cb7-12" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb7-13"><a href="#cb7-13" aria-hidden="true" tabindex="-1"></a> <span class="fu">Solution</span>(<span class="fu">y</span>.(t), <span class="fu">v</span>.(t), t, prob, alg)</span>
<span id="cb7-14"><a href="#cb7-14" aria-hidden="true" tabindex="-1"></a><span class="kw">end</span></span>
<span id="cb7-15"><a href="#cb7-15" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb7-16"><a href="#cb7-16" aria-hidden="true" tabindex="-1"></a><span class="kw">function</span> <span class="fu">solve</span>(prob<span class="op">::</span><span class="dt">Problem</span>, alg<span class="op">::</span><span class="dt">EulerMethod</span>)</span>
<span id="cb7-17"><a href="#cb7-17" aria-hidden="true" tabindex="-1"></a> g, y0, v0, tspan <span class="op">=</span> prob.g, prob.y0, prob.v0, prob.tspan</span>
<span id="cb7-18"><a href="#cb7-18" aria-hidden="true" tabindex="-1"></a> dt <span class="op">=</span> alg.dt</span>
<span id="cb7-19"><a href="#cb7-19" aria-hidden="true" tabindex="-1"></a> t0, t1 <span class="op">=</span> tspan</span>
<span id="cb7-20"><a href="#cb7-20" aria-hidden="true" tabindex="-1"></a> t <span class="op">=</span> <span class="fu">range</span>(t0, t1 <span class="op">+</span> dt<span class="op">/</span><span class="fl">2</span>; step <span class="op">=</span> dt)</span>
<span id="cb7-21"><a href="#cb7-21" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb7-22"><a href="#cb7-22" aria-hidden="true" tabindex="-1"></a> n <span class="op">=</span> <span class="fu">length</span>(t)</span>
<span id="cb7-23"><a href="#cb7-23" aria-hidden="true" tabindex="-1"></a> y <span class="op">=</span> <span class="fu">Vector</span><span class="dt">{typeof(y0)}</span>(<span class="cn">undef</span>, n)</span>
<span id="cb7-24"><a href="#cb7-24" aria-hidden="true" tabindex="-1"></a> v <span class="op">=</span> <span class="fu">Vector</span><span class="dt">{typeof(v0)}</span>(<span class="cn">undef</span>, n)</span>
<span id="cb7-25"><a href="#cb7-25" aria-hidden="true" tabindex="-1"></a> y[<span class="fl">1</span>] <span class="op">=</span> y0</span>
<span id="cb7-26"><a href="#cb7-26" aria-hidden="true" tabindex="-1"></a> v[<span class="fl">1</span>] <span class="op">=</span> v0</span>
<span id="cb7-27"><a href="#cb7-27" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb7-28"><a href="#cb7-28" aria-hidden="true" tabindex="-1"></a> <span class="cf">for</span> i <span class="kw">in</span> <span class="fl">1</span><span class="op">:</span>n<span class="op">-</span><span class="fl">1</span></span>
<span id="cb7-29"><a href="#cb7-29" aria-hidden="true" tabindex="-1"></a> v[i<span class="op">+</span><span class="fl">1</span>] <span class="op">=</span> v[i] <span class="op">-</span> g<span class="op">*</span>dt <span class="co"># F*h step of Euler</span></span>
<span id="cb7-30"><a href="#cb7-30" aria-hidden="true" tabindex="-1"></a> y[i<span class="op">+</span><span class="fl">1</span>] <span class="op">=</span> y[i] <span class="op">+</span> v[i]<span class="op">*</span>dt <span class="co"># F*h step of Euler</span></span>
<span id="cb7-31"><a href="#cb7-31" aria-hidden="true" tabindex="-1"></a> <span class="cf">end</span></span>
<span id="cb7-32"><a href="#cb7-32" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb7-33"><a href="#cb7-33" aria-hidden="true" tabindex="-1"></a> <span class="fu">Solution</span>(y, v, t, prob, alg)</span>
<span id="cb7-34"><a href="#cb7-34" aria-hidden="true" tabindex="-1"></a><span class="kw">end</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-display" data-execution_count="8">
<pre><code>solve (generic function with 3 methods)</code></pre>
</div>
</div>
<p>The post has a more elegant means to unpack the parameters from the structures, but for each of the above, the parameters are unpacked, and then the corresponding algorithm employed. As of version <code>v1.7</code> of <code>Julia</code>, the syntax <code>(;g,y0,v0,tspan) = prob</code> could also be employed.</p>
<p>The exact formulas, <code>y(t) = y0 + v0*(t - t0) - g*(t - t0)^2/2</code> and <code>v(t) = v0 - g*(t - t0)</code>, follow from well-known physics formulas. Each answer is wrapped in a <code>Solution</code> type so that the answers found can be easily extracted in a uniform manner.</p>
<p>For example, plots of each can be obtained through:</p>
<div class="cell" data-execution_count="8">
<div class="sourceCode cell-code" id="cb9"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb9-1"><a href="#cb9-1" aria-hidden="true" tabindex="-1"></a>earth <span class="op">=</span> <span class="fu">Problem</span>()</span>
<span id="cb9-2"><a href="#cb9-2" aria-hidden="true" tabindex="-1"></a>sol_euler <span class="op">=</span> <span class="fu">solve</span>(earth)</span>
<span id="cb9-3"><a href="#cb9-3" aria-hidden="true" tabindex="-1"></a>sol_exact <span class="op">=</span> <span class="fu">solve</span>(earth, <span class="fu">ExactFormula</span>())</span>
<span id="cb9-4"><a href="#cb9-4" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb9-5"><a href="#cb9-5" aria-hidden="true" tabindex="-1"></a><span class="fu">plot</span>(sol_euler.t, sol_euler.y;</span>
<span id="cb9-6"><a href="#cb9-6" aria-hidden="true" tabindex="-1"></a> label<span class="op">=</span><span class="st">"Euler's method (dt = </span><span class="sc">$</span>(sol_euler.alg.dt)<span class="st">)"</span>, ls<span class="op">=:</span>auto)</span>
<span id="cb9-7"><a href="#cb9-7" aria-hidden="true" tabindex="-1"></a><span class="fu">plot!</span>(sol_exact.t, sol_exact.y; label<span class="op">=</span><span class="st">"exact solution"</span>, ls<span class="op">=:</span>auto)</span>
<span id="cb9-8"><a href="#cb9-8" aria-hidden="true" tabindex="-1"></a><span class="fu">title!</span>(<span class="st">"On the Earth"</span>; xlabel<span class="op">=</span><span class="st">"t"</span>, legend<span class="op">=:</span>bottomleft)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-display" data-execution_count="9">
<p><img src="solve_files/figure-html/cell-9-output-1.svg" class="img-fluid"></p>
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<p>Following the post, since the time step <code>dt = 0.1</code> is not small enough, the error of the Euler method is rather large. Next we change the algorithm parameter, <code>dt</code>, to be smaller:</p>
<div class="cell" data-execution_count="9">
<div class="sourceCode cell-code" id="cb10"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb10-1"><a href="#cb10-1" aria-hidden="true" tabindex="-1"></a>earth₂ <span class="op">=</span> <span class="fu">Problem</span>()</span>
<span id="cb10-2"><a href="#cb10-2" aria-hidden="true" tabindex="-1"></a>sol_euler₂ <span class="op">=</span> <span class="fu">solve</span>(earth₂, <span class="fu">EulerMethod</span>(dt <span class="op">=</span> <span class="fl">0.01</span>))</span>
<span id="cb10-3"><a href="#cb10-3" aria-hidden="true" tabindex="-1"></a>sol_exact₂ <span class="op">=</span> <span class="fu">solve</span>(earth₂, <span class="fu">ExactFormula</span>())</span>
<span id="cb10-4"><a href="#cb10-4" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb10-5"><a href="#cb10-5" aria-hidden="true" tabindex="-1"></a><span class="fu">plot</span>(sol_euler₂.t, sol_euler₂.y;</span>
<span id="cb10-6"><a href="#cb10-6" aria-hidden="true" tabindex="-1"></a> label<span class="op">=</span><span class="st">"Euler's method (dt = </span><span class="sc">$</span>(sol_euler₂.alg.dt)<span class="st">)"</span>, ls<span class="op">=:</span>auto)</span>
<span id="cb10-7"><a href="#cb10-7" aria-hidden="true" tabindex="-1"></a><span class="fu">plot!</span>(sol_exact₂.t, sol_exact₂.y; label<span class="op">=</span><span class="st">"exact solution"</span>, ls<span class="op">=:</span>auto)</span>
<span id="cb10-8"><a href="#cb10-8" aria-hidden="true" tabindex="-1"></a><span class="fu">title!</span>(<span class="st">"On the Earth"</span>; xlabel<span class="op">=</span><span class="st">"t"</span>, legend<span class="op">=:</span>bottomleft)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-display" data-execution_count="10">
<p><img src="solve_files/figure-html/cell-10-output-1.svg" class="img-fluid"></p>
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<p>It is worth noting that only the first line is modified, and only the method requires modification.</p>
<p>Were the moon to be considered, the gravitational constant would need adjustment. This parameter is part of the problem, not the solution algorithm.</p>
<p>Such adjustments are made by passing different values to the <code>Problem</code> constructor:</p>
<div class="cell" data-execution_count="10">
<div class="sourceCode cell-code" id="cb11"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb11-1"><a href="#cb11-1" aria-hidden="true" tabindex="-1"></a>moon <span class="op">=</span> <span class="fu">Problem</span>(g <span class="op">=</span> <span class="fl">1.62</span>, tspan <span class="op">=</span> (<span class="fl">0.0</span>, <span class="fl">40.0</span>))</span>
<span id="cb11-2"><a href="#cb11-2" aria-hidden="true" tabindex="-1"></a>sol_eulerₘ <span class="op">=</span> <span class="fu">solve</span>(moon)</span>
<span id="cb11-3"><a href="#cb11-3" aria-hidden="true" tabindex="-1"></a>sol_exactₘ <span class="op">=</span> <span class="fu">solve</span>(moon, <span class="fu">ExactFormula</span>(dt <span class="op">=</span> sol_euler.alg.dt))</span>
<span id="cb11-4"><a href="#cb11-4" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb11-5"><a href="#cb11-5" aria-hidden="true" tabindex="-1"></a><span class="fu">plot</span>(sol_eulerₘ.t, sol_eulerₘ.y;</span>
<span id="cb11-6"><a href="#cb11-6" aria-hidden="true" tabindex="-1"></a> label<span class="op">=</span><span class="st">"Euler's method (dt = </span><span class="sc">$</span>(sol_eulerₘ.alg.dt)<span class="st">)"</span>, ls<span class="op">=:</span>auto)</span>
<span id="cb11-7"><a href="#cb11-7" aria-hidden="true" tabindex="-1"></a><span class="fu">plot!</span>(sol_exactₘ.t, sol_exactₘ.y; label<span class="op">=</span><span class="st">"exact solution"</span>, ls<span class="op">=:</span>auto)</span>
<span id="cb11-8"><a href="#cb11-8" aria-hidden="true" tabindex="-1"></a><span class="fu">title!</span>(<span class="st">"On the Moon"</span>; xlabel<span class="op">=</span><span class="st">"t"</span>, legend<span class="op">=:</span>bottomleft)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<p><img src="solve_files/figure-html/cell-11-output-1.svg" class="img-fluid"></p>
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<p>The code above also adjusts the time span in addition to the graviational constant. The algorithm for exact formula is set to use the <code>dt</code> value used in the <code>euler</code> formula, for easier comparison. Otherwise, outside of the labels, the patterns are the same. Only those things that need changing are changed, the rest comes from defaults.</p>
<p>The above shows the benefits of using a common interface. Next, the post illustrates how <em>other</em> authors could extend this code, simply by adding a <em>new</em> <code>solve</code> method. For example,</p>
<div class="cell" data-execution_count="11">
<div class="sourceCode cell-code" id="cb12"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb12-1"><a href="#cb12-1" aria-hidden="true" tabindex="-1"></a><span class="kw">struct</span> Symplectic2ndOrder{T}</span>
<span id="cb12-2"><a href="#cb12-2" aria-hidden="true" tabindex="-1"></a> dt<span class="op">::</span><span class="dt">T</span></span>
<span id="cb12-3"><a href="#cb12-3" aria-hidden="true" tabindex="-1"></a><span class="kw">end</span></span>
<span id="cb12-4"><a href="#cb12-4" aria-hidden="true" tabindex="-1"></a><span class="fu">Symplectic2ndOrder</span>(;dt<span class="op">=</span><span class="fl">0.1</span>) <span class="op">=</span> <span class="fu">Symplectic2ndOrder</span>(dt)</span>
<span id="cb12-5"><a href="#cb12-5" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb12-6"><a href="#cb12-6" aria-hidden="true" tabindex="-1"></a><span class="kw">function</span> <span class="fu">solve</span>(prob<span class="op">::</span><span class="dt">Problem</span>, alg<span class="op">::</span><span class="dt">Symplectic2ndOrder</span>)</span>
<span id="cb12-7"><a href="#cb12-7" aria-hidden="true" tabindex="-1"></a> g, y0, v0, tspan <span class="op">=</span> prob.g, prob.y0, prob.v0, prob.tspan</span>
<span id="cb12-8"><a href="#cb12-8" aria-hidden="true" tabindex="-1"></a> dt <span class="op">=</span> alg.dt</span>
<span id="cb12-9"><a href="#cb12-9" aria-hidden="true" tabindex="-1"></a> t0, t1 <span class="op">=</span> tspan</span>
<span id="cb12-10"><a href="#cb12-10" aria-hidden="true" tabindex="-1"></a> t <span class="op">=</span> <span class="fu">range</span>(t0, t1 <span class="op">+</span> dt<span class="op">/</span><span class="fl">2</span>; step <span class="op">=</span> dt)</span>
<span id="cb12-11"><a href="#cb12-11" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb12-12"><a href="#cb12-12" aria-hidden="true" tabindex="-1"></a> n <span class="op">=</span> <span class="fu">length</span>(t)</span>
<span id="cb12-13"><a href="#cb12-13" aria-hidden="true" tabindex="-1"></a> y <span class="op">=</span> <span class="fu">Vector</span><span class="dt">{typeof(y0)}</span>(<span class="cn">undef</span>, n)</span>
<span id="cb12-14"><a href="#cb12-14" aria-hidden="true" tabindex="-1"></a> v <span class="op">=</span> <span class="fu">Vector</span><span class="dt">{typeof(v0)}</span>(<span class="cn">undef</span>, n)</span>
<span id="cb12-15"><a href="#cb12-15" aria-hidden="true" tabindex="-1"></a> y[<span class="fl">1</span>] <span class="op">=</span> y0</span>
<span id="cb12-16"><a href="#cb12-16" aria-hidden="true" tabindex="-1"></a> v[<span class="fl">1</span>] <span class="op">=</span> v0</span>
<span id="cb12-17"><a href="#cb12-17" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb12-18"><a href="#cb12-18" aria-hidden="true" tabindex="-1"></a> <span class="cf">for</span> i <span class="kw">in</span> <span class="fl">1</span><span class="op">:</span>n<span class="op">-</span><span class="fl">1</span></span>
<span id="cb12-19"><a href="#cb12-19" aria-hidden="true" tabindex="-1"></a> ytmp <span class="op">=</span> y[i] <span class="op">+</span> v[i]<span class="op">*</span>dt<span class="op">/</span><span class="fl">2</span></span>
<span id="cb12-20"><a href="#cb12-20" aria-hidden="true" tabindex="-1"></a> v[i<span class="op">+</span><span class="fl">1</span>] <span class="op">=</span> v[i] <span class="op">-</span> g<span class="op">*</span>dt</span>
<span id="cb12-21"><a href="#cb12-21" aria-hidden="true" tabindex="-1"></a> y[i<span class="op">+</span><span class="fl">1</span>] <span class="op">=</span> ytmp <span class="op">+</span> v[i<span class="op">+</span><span class="fl">1</span>]<span class="op">*</span>dt<span class="op">/</span><span class="fl">2</span></span>
<span id="cb12-22"><a href="#cb12-22" aria-hidden="true" tabindex="-1"></a> <span class="cf">end</span></span>
<span id="cb12-23"><a href="#cb12-23" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb12-24"><a href="#cb12-24" aria-hidden="true" tabindex="-1"></a> <span class="fu">Solution</span>(y, v, t, prob, alg)</span>
<span id="cb12-25"><a href="#cb12-25" aria-hidden="true" tabindex="-1"></a><span class="kw">end</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-display" data-execution_count="12">
<pre><code>solve (generic function with 4 methods)</code></pre>
</div>
</div>
<p>Had the two prior methods been in a package, the other user could still extend the interface, as above, with just a slight standard modification.</p>
<p>The same approach works for this new type:</p>
<div class="cell" data-execution_count="12">
<div class="sourceCode cell-code" id="cb14"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb14-1"><a href="#cb14-1" aria-hidden="true" tabindex="-1"></a>earth₃ <span class="op">=</span> <span class="fu">Problem</span>()</span>
<span id="cb14-2"><a href="#cb14-2" aria-hidden="true" tabindex="-1"></a>sol_sympl₃ <span class="op">=</span> <span class="fu">solve</span>(earth₃, <span class="fu">Symplectic2ndOrder</span>(dt <span class="op">=</span> <span class="fl">2.0</span>))</span>
<span id="cb14-3"><a href="#cb14-3" aria-hidden="true" tabindex="-1"></a>sol_exact₃ <span class="op">=</span> <span class="fu">solve</span>(earth₃, <span class="fu">ExactFormula</span>())</span>
<span id="cb14-4"><a href="#cb14-4" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb14-5"><a href="#cb14-5" aria-hidden="true" tabindex="-1"></a><span class="fu">plot</span>(sol_sympl₃.t, sol_sympl₃.y; label<span class="op">=</span><span class="st">"2nd order symplectic (dt = </span><span class="sc">$</span>(sol_sympl₃.alg.dt)<span class="st">)"</span>, ls<span class="op">=:</span>auto)</span>
<span id="cb14-6"><a href="#cb14-6" aria-hidden="true" tabindex="-1"></a><span class="fu">plot!</span>(sol_exact₃.t, sol_exact₃.y; label<span class="op">=</span><span class="st">"exact solution"</span>, ls<span class="op">=:</span>auto)</span>
<span id="cb14-7"><a href="#cb14-7" aria-hidden="true" tabindex="-1"></a><span class="fu">title!</span>(<span class="st">"On the Earth"</span>; xlabel<span class="op">=</span><span class="st">"t"</span>, legend<span class="op">=:</span>bottomleft)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-display" data-execution_count="13">
<p><img src="solve_files/figure-html/cell-13-output-1.svg" class="img-fluid"></p>
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<p>Finally, the author of the post shows how the interface can compose with other packages in the <code>Julia</code> package ecosystem. This example uses the external package <code>MonteCarloMeasurements</code> which plots the behavior of the system for perturbations of the initial value:</p>
<div class="cell" data-execution_count="13">
<div class="sourceCode cell-code" id="cb15"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb15-1"><a href="#cb15-1" aria-hidden="true" tabindex="-1"></a>earth₄ <span class="op">=</span> <span class="fu">Problem</span>(y0 <span class="op">=</span> <span class="fl">0.0</span> <span class="op">±</span> <span class="fl">0.0</span>, v0 <span class="op">=</span> <span class="fl">30.0</span> <span class="op">±</span> <span class="fl">1.0</span>)</span>
<span id="cb15-2"><a href="#cb15-2" aria-hidden="true" tabindex="-1"></a>sol_euler₄ <span class="op">=</span> <span class="fu">solve</span>(earth₄)</span>
<span id="cb15-3"><a href="#cb15-3" aria-hidden="true" tabindex="-1"></a>sol_sympl₄ <span class="op">=</span> <span class="fu">solve</span>(earth₄, <span class="fu">Symplectic2ndOrder</span>(dt <span class="op">=</span> <span class="fl">2.0</span>))</span>
<span id="cb15-4"><a href="#cb15-4" aria-hidden="true" tabindex="-1"></a>sol_exact₄ <span class="op">=</span> <span class="fu">solve</span>(earth₄, <span class="fu">ExactFormula</span>())</span>
<span id="cb15-5"><a href="#cb15-5" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb15-6"><a href="#cb15-6" aria-hidden="true" tabindex="-1"></a>ylim <span class="op">=</span> (<span class="op">-</span><span class="fl">100</span>, <span class="fl">60</span>)</span>
<span id="cb15-7"><a href="#cb15-7" aria-hidden="true" tabindex="-1"></a>P <span class="op">=</span> <span class="fu">plot</span>(sol_euler₄.t, sol_euler₄.y;</span>
<span id="cb15-8"><a href="#cb15-8" aria-hidden="true" tabindex="-1"></a> label<span class="op">=</span><span class="st">"Euler's method (dt = </span><span class="sc">$</span>(sol_euler₄.alg.dt)<span class="st">)"</span>, ls<span class="op">=:</span>auto)</span>
<span id="cb15-9"><a href="#cb15-9" aria-hidden="true" tabindex="-1"></a><span class="fu">title!</span>(<span class="st">"On the Earth"</span>; xlabel<span class="op">=</span><span class="st">"t"</span>, legend<span class="op">=:</span>bottomleft, ylim)</span>
<span id="cb15-10"><a href="#cb15-10" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb15-11"><a href="#cb15-11" aria-hidden="true" tabindex="-1"></a>Q <span class="op">=</span> <span class="fu">plot</span>(sol_sympl₄.t, sol_sympl₄.y;</span>
<span id="cb15-12"><a href="#cb15-12" aria-hidden="true" tabindex="-1"></a> label<span class="op">=</span><span class="st">"2nd order symplectic (dt = </span><span class="sc">$</span>(sol_sympl₄.alg.dt)<span class="st">)"</span>, ls<span class="op">=:</span>auto)</span>
<span id="cb15-13"><a href="#cb15-13" aria-hidden="true" tabindex="-1"></a><span class="fu">title!</span>(<span class="st">"On the Earth"</span>; xlabel<span class="op">=</span><span class="st">"t"</span>, legend<span class="op">=:</span>bottomleft, ylim)</span>
<span id="cb15-14"><a href="#cb15-14" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb15-15"><a href="#cb15-15" aria-hidden="true" tabindex="-1"></a>R <span class="op">=</span> <span class="fu">plot</span>(sol_exact₄.t, sol_exact₄.y; label<span class="op">=</span><span class="st">"exact solution"</span>, ls<span class="op">=:</span>auto)</span>
<span id="cb15-16"><a href="#cb15-16" aria-hidden="true" tabindex="-1"></a><span class="fu">title!</span>(<span class="st">"On the Earth"</span>; xlabel<span class="op">=</span><span class="st">"t"</span>, legend<span class="op">=:</span>bottomleft, ylim)</span>
<span id="cb15-17"><a href="#cb15-17" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb15-18"><a href="#cb15-18" aria-hidden="true" tabindex="-1"></a><span class="fu">plot</span>(P, Q, R; size<span class="op">=</span>(<span class="fl">720</span>, <span class="fl">600</span>))</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-display" data-execution_count="14">
<p><img src="solve_files/figure-html/cell-14-output-1.svg" class="img-fluid"></p>
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<p>The only change was in the problem, <code>Problem(y0 = 0.0 ± 0.0, v0 = 30.0 ± 1.0)</code>, where a different number type is used which accounts for uncertainty. The rest follows the same pattern.</p>
<p>This example, shows the flexibility of the problem-algorithm-solver pattern while maintaining a consistent pattern for execution.</p>
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