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<meta charset="utf-8" />
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<title>Splitting Methods for the time-dependent convection diffusion equation &#8212; Introduction to Scientific Computing</title>
<title>16. Splitting Methods for the time-dependent convection diffusion equation &#8212; Introduction to Scientific Computing</title>



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Introduction to Scientific Computing
An Interactive Introduction to the Finite Element Method
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<li class="toctree-l1"><a class="reference internal" href="../primal/boundary_conditions.html">2. Boundary Conditions</a></li>
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<li class="toctree-l1"><a class="reference internal" href="elliptic.html">15. Hybrid DG for elliptic equations</a></li>
<li class="toctree-l1 current active"><a class="current reference internal" href="#">16. Splitting Methods for the time-dependent convection diffusion equation</a></li>
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<article class="bd-article">

<section class="tex2jax_ignore mathjax_ignore" id="splitting-methods-for-the-time-dependent-convection-diffusion-equation">
<h1>Splitting Methods for the time-dependent convection diffusion equation<a class="headerlink" href="#splitting-methods-for-the-time-dependent-convection-diffusion-equation" title="Link to this heading">#</a></h1>
<h1><span class="section-number">16. </span>Splitting Methods for the time-dependent convection diffusion equation<a class="headerlink" href="#splitting-methods-for-the-time-dependent-convection-diffusion-equation" title="Link to this heading">#</a></h1>
<p><span class="math notranslate nohighlight">\(\DeclareMathOperator{\opdiv}{div}\)</span></p>
<p>We now want to solve a time-dependent convection diffusion equation:</p>
<div class="math notranslate nohighlight">
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</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>&lt;ngsolve.comp.LinearForm at 0x1169ba630&gt;
</pre></div>
</div>
</div>
</div>
<div class="cell docutils container">
<div class="cell_input docutils container">
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</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<script type="application/vnd.jupyter.widget-view+json">{"version_major": 2, "version_minor": 0, "model_id": "4c9a4999127847e6a09c8ac8f920a567"}</script><div class="output traceback highlight-ipythontb notranslate"><div class="highlight"><pre><span></span><span class="gt">---------------------------------------------------------------------------</span>
<span class="ne">KeyboardInterrupt</span><span class="g g-Whitespace"> </span>Traceback (most recent call last)
<span class="n">Cell</span> <span class="n">In</span><span class="p">[</span><span class="mi">4</span><span class="p">],</span> <span class="n">line</span> <span class="mi">15</span>
<span class="g g-Whitespace"> </span><span class="mi">13</span> <span class="k">while</span> <span class="n">t</span> <span class="o">&lt;</span> <span class="n">tend</span><span class="p">:</span>
<span class="g g-Whitespace"> </span><span class="mi">14</span> <span class="n">t</span> <span class="o">+=</span> <span class="n">tau</span>
<span class="ne">---&gt; </span><span class="mi">15</span> <span class="n">aconv</span><span class="o">.</span><span class="n">Apply</span><span class="p">(</span><span class="n">gfu</span><span class="o">.</span><span class="n">vec</span><span class="p">,</span> <span class="n">convu</span><span class="p">)</span>
<span class="g g-Whitespace"> </span><span class="mi">16</span> <span class="n">r</span><span class="o">.</span><span class="n">data</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">vec</span> <span class="o">-</span> <span class="n">convu</span> <span class="o">-</span> <span class="n">adiff</span><span class="o">.</span><span class="n">mat</span> <span class="o">*</span> <span class="n">gfu</span><span class="o">.</span><span class="n">vec</span>
<span class="g g-Whitespace"> </span><span class="mi">17</span> <span class="n">w</span><span class="o">.</span><span class="n">data</span> <span class="o">=</span> <span class="n">inv</span> <span class="o">*</span> <span class="n">r</span>

<span class="ne">KeyboardInterrupt</span>:
</pre></div>
</div>
</div>
</div>
<p>For higher order IMplicit-EXplicit (IMEX) methods see:</p>
<p><em>Implicit-explicit Runge-Kutta methods for time-dependent pdes</em> by Uri M. Ascher, Steven J. Ruuth, and Raymond J. Spiteri, Applied Numerical Mathematics, 25, 1997</p>
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Expand All @@ -182,6 +182,38 @@
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