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58 | 58 | <li class="toctree-l3"><a class="reference internal" href="#eigenvalue-solver">Eigenvalue Solver</a></li> |
59 | 59 | </ul> |
60 | 60 | </li> |
| 61 | +<li class="toctree-l2"><a class="reference internal" href="#computational-density-and-comparison">Computational Density and Comparison</a><ul> |
| 62 | +<li class="toctree-l3"><a class="reference internal" href="#general-tips">General Tips</a></li> |
| 63 | +</ul> |
| 64 | +</li> |
61 | 65 | </ul> |
62 | 66 | </li> |
63 | | -<li class="toctree-l1"><a class="reference internal" href="#computational-density-and-comparison">Computational Density and Comparison</a></li> |
64 | 67 | <li class="toctree-l1"><a class="reference internal" href="4_input.html">Input Data Files</a></li> |
65 | 68 | <li class="toctree-l1"><a class="reference internal" href="5_gui.html">Graphical User Interface (GUI)</a></li> |
66 | 69 | <li class="toctree-l1"><a class="reference internal" href="6_inputdeck.html">Input Deck for Solver Parameters</a></li> |
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109 | 112 | <section id="time-integration"> |
110 | 113 | <span id="marching"></span><h3>Time Integration<a class="headerlink" href="#time-integration" title="Link to this heading"></a></h3> |
111 | 114 | <p><strong>Explicit Euler Method</strong></p> |
112 | | -<p>The first-order explicit Euler scheme advances the position by sampling the velocity at the beginning of the time step:</p> |
| 115 | +<p>The first-order explicit Euler scheme <code class="docutils literal notranslate"><span class="pre">Euler</span></code> advances the position by sampling the velocity at the beginning of the time step:</p> |
113 | 116 | <div class="math notranslate nohighlight"> |
114 | 117 | \[\begin{split}\mathbf{u}_n = \mathbf{u}(\mathbf{x}_n,t_n),\\ |
115 | 118 | \mathbf{x}_{n+1} = \mathbf{x}_n + \sigma\,\Delta t\,\mathbf{u}_n.\end{split}\]</div> |
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340 | 343 | <span id="eigen"></span><h3>Eigenvalue Solver<a class="headerlink" href="#eigenvalue-solver" title="Link to this heading"></a></h3> |
341 | 344 | </section> |
342 | 345 | </section> |
343 | | -</section> |
344 | 346 | <section id="computational-density-and-comparison"> |
345 | | -<span id="numcompare"></span><h1>Computational Density and Comparison<a class="headerlink" href="#computational-density-and-comparison" title="Link to this heading"></a></h1> |
346 | | -<p id="numtips">As for your reference, and configured as defaults, the <em>Berkeley LCS Tutorials</em> used <code class="docutils literal notranslate"><span class="pre">RK4</span></code> for advection. |
| 347 | +<span id="numcompare"></span><h2>Computational Density and Comparison<a class="headerlink" href="#computational-density-and-comparison" title="Link to this heading"></a></h2> |
| 348 | +<section id="general-tips"> |
| 349 | +<span id="numtips"></span><h3>General Tips<a class="headerlink" href="#general-tips" title="Link to this heading"></a></h3> |
| 350 | +<p>As for your reference, and configured as defaults, the <em>Berkeley LCS Tutorials</em> used <code class="docutils literal notranslate"><span class="pre">RK4</span></code> for advection. |
347 | 351 | The velocity fields were interpolated with <code class="docutils literal notranslate"><span class="pre">tricubic-FL</span></code> by them, originating from <a class="reference internal" href="9_references.html#lekien2005" id="id2"><span>[Lekien2005]</span></a>, which has higher performance by solving a 64×64 linear system using the function values, gradients, and mixed partial derivatives at its eight corners, which is in future development plan for <code class="docutils literal notranslate"><span class="pre">Py3DFTLE</span></code> with high priority. |
348 | 352 | Although not detailed, <code class="docutils literal notranslate"><span class="pre">grad_order=2</span></code> was employed by them from the equation, supposing the mesh is sufficiently refined.</p> |
349 | 353 | <p>Please always notice that, although providing much better numerical precision and looks cool in papers, high-order methods could be resource-consuming, even several hundred times.</p> |
| 354 | +</section> |
| 355 | +</section> |
350 | 356 | </section> |
351 | 357 |
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352 | 358 |
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