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‎.doctrees/3_numerical.doctree‎

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‎.doctrees/9_references.doctree‎

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‎.doctrees/environment.pickle‎

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‎3_numerical.html‎

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<p>The classical RK4 method achieves fourth-order accuracy via four slope evaluations at intermediate points:</p>
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<p>This yields a global error of order <span class="math notranslate nohighlight">\(O(\Delta t^4)\)</span> with four velocity evaluations per step.</p>
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<p><strong>Sixth-Order Runge-Kutta (RK6)</strong></p>
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<p>The seven-stage scheme <code class="docutils literal notranslate"><span class="pre">ERK6(7)</span></code> uses non-uniform weights to attain global <span class="math notranslate nohighlight">\(O(\Delta t^6)\)</span> accuracy. This method originates from <a class="reference internal" href="9_references.html#butcher" id="id1"><span>[Butcher]</span></a>.
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<p>The seven-stage scheme <code class="docutils literal notranslate"><span class="pre">ERK6(7)</span></code> uses non-uniform weights to attain global <span class="math notranslate nohighlight">\(O(\Delta t^6)\)</span> accuracy, originating from <a class="reference internal" href="9_references.html#butcher1964" id="id1"><span>[Butcher1964]</span></a>.
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As for the coefficients for <code class="docutils literal notranslate"><span class="pre">RK6</span></code> are more complex to write into equations, the Butcher table is given as follows.</p>
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<table class="docutils align-default">
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<thead>
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</tr>
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</tbody>
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</table>
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<p>In our computation, the up symbol side is applied, in other words, <code class="docutils literal notranslate"><span class="pre">±</span></code> represents <code class="docutils literal notranslate"><span class="pre">+</span></code>, taking <span class="math notranslate nohighlight">\(\lambda=+\sqrt{5}\)</span>.</p>
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<p>In our computation, the up symbol side is applied, in other words, <code class="docutils literal notranslate"><span class="pre">±</span></code> represents <code class="docutils literal notranslate"><span class="pre">+</span></code>, taking <span class="math notranslate nohighlight">\(\lambda=+\sqrt{5}\)</span>. With 15 digis are kept, the Butcher table used by the author is shown in the following table.</p>
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<table class="docutils align-default">
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<thead>
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<tr class="row-odd"><th class="head"><p><span class="math notranslate nohighlight">\(c_i\)</span></p></th>
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<th class="head"><p><span class="math notranslate nohighlight">\(a_{i1}\)</span></p></th>
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<th class="head"><p><span class="math notranslate nohighlight">\(a_{i2}\)</span></p></th>
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<th class="head"><p><span class="math notranslate nohighlight">\(a_{i3}\)</span></p></th>
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<th class="head"><p><span class="math notranslate nohighlight">\(a_{i4}\)</span></p></th>
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<th class="head"><p><span class="math notranslate nohighlight">\(a_{i5}\)</span></p></th>
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<th class="head"><p><span class="math notranslate nohighlight">\(a_{i6}\)</span></p></th>
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<th class="head"><p><span class="math notranslate nohighlight">\(a_{i7}\)</span></p></th>
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</tr>
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</thead>
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<tbody>
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<tr class="row-even"><td><p>0</p></td>
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<td><p>0</p></td>
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<td><p>0</p></td>
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<td><p>0</p></td>
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<td><p>0</p></td>
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<td><p>0</p></td>
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<td><p>0</p></td>
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<td><p>0</p></td>
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</tr>
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<tr class="row-odd"><td><p>0.276393202250021</p></td>
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<td><p>0.276393202250021</p></td>
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<td><p>0</p></td>
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<td><p>0</p></td>
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<td><p>0</p></td>
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<td><p>0</p></td>
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<td><p>0</p></td>
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<td><p>0</p></td>
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</tr>
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<tr class="row-even"><td><p>0.723606797749979</p></td>
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<td><p>-0.223606797749979</p></td>
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<td><p>0.947213595499958</p></td>
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<td><p>0</p></td>
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<td><p>0</p></td>
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<td><p>0</p></td>
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<td><p>0</p></td>
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<td><p>0</p></td>
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</tr>
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<tr class="row-odd"><td><p>0.276393202250021</p></td>
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<td><p>0.0326237921249264</p></td>
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<td><p>0.309016994374947</p></td>
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<td><p>-0.0652475842498529</p></td>
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<td><p>0</p></td>
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<td><p>0</p></td>
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<td><p>0</p></td>
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<td><p>0</p></td>
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</tr>
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<tr class="row-even"><td><p>0.723606797749979</p></td>
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<td><p>0.0460655337083368</p></td>
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<td><p>0</p></td>
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<td><p>0.166666666666667</p></td>
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<td><p>0.510874597374975</p></td>
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<td><p>0</p></td>
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<td><p>0</p></td>
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<td><p>0</p></td>
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</tr>
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<tr class="row-odd"><td><p>0.276393202250021</p></td>
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<td><p>0.12060113295833</p></td>
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<td><p>0</p></td>
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<td><p>-0.181694990624912</p></td>
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<td><p>0.166666666666667</p></td>
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<td><p>0.170820393249937</p></td>
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<td><p>0</p></td>
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<td><p>0</p></td>
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</tr>
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<tr class="row-even"><td><p>1</p></td>
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<td><p>0.166666666666667</p></td>
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<td><p>0</p></td>
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<td><p>0.0751416197912285</p></td>
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<td><p>-3.38770632020821</p></td>
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<td><p>0.52786404500042</p></td>
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<td><p>3.61803398874989</p></td>
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<td><p>0</p></td>
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</tr>
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<tr class="row-odd"><td><p>b_i</p></td>
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<td><p>0.0833333333333333</p></td>
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<td><p>0</p></td>
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<td><p>0</p></td>
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<td><p>0</p></td>
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<td><p>0.416666666666667</p></td>
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<td><p>0.416666666666667</p></td>
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<td><p>0.0833333333333333</p></td>
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</tr>
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</tbody>
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</table>
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</section>
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</section>
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<section id="ftle-computation">

‎9_references.html‎

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<span class="label"><span class="fn-bracket">[</span>Lekien2005<span class="fn-bracket">]</span></span>
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<p>Lekien, F., Marsden, J. “Tricubic interpolation in three dimensions,” <em>International Journal for Numerical Methods in Engineering</em> <strong>63</strong> (2005).</p>
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</div>
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<div class="citation" id="butcher" role="doc-biblioentry">
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<span class="label"><span class="fn-bracket">[</span>Butcher<span class="fn-bracket">]</span></span>
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<div class="citation" id="butcher1964" role="doc-biblioentry">
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<span class="label"><span class="fn-bracket">[</span>Butcher1964<span class="fn-bracket">]</span></span>
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<p>Butcher J. C. “On Runge-Kutta processes of high order.” <em>Journal of the Australian Mathematical Society</em> <strong>4(2)</strong> (1964). doi:10.1017/S1446788700023387</p>
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</div>
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</div>

‎_sources/3_numerical.rst.txt‎

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**Sixth-Order Runge-Kutta (RK6)**
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The seven-stage scheme ``ERK6(7)`` uses non-uniform weights to attain global :math:`O(\Delta t^6)` accuracy. This method originates from [Butcher]_.
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The seven-stage scheme ``ERK6(7)`` uses non-uniform weights to attain global :math:`O(\Delta t^6)` accuracy, originating from [Butcher1964]_.
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As for the coefficients for ``RK6`` are more complex to write into equations, the Butcher table is given as follows.
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+-----------------------------+-------------------------------+----------------------------+---------------------------------+------------------------------+-----------------------------+--------------------------+-------------------------+
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| :math:`b_i` | :math:`1/12` | :math:`0` | :math:`0` | :math:`0` | :math:`5/12` | :math:`5/12` | :math:`1/12` |
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+-----------------------------+-------------------------------+----------------------------+---------------------------------+------------------------------+-----------------------------+--------------------------+-------------------------+
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In our computation, the up symbol side is applied, in other words, ``±`` represents ``+``, taking :math:`\lambda=+\sqrt{5}`.
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In our computation, the up symbol side is applied, in other words, ``±`` represents ``+``, taking :math:`\lambda=+\sqrt{5}`. With 15 digis are kept, the Butcher table used by the author is shown in the following table.
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+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+
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| :math:`c_i` | :math:`a_{i1}` | :math:`a_{i2}` | :math:`a_{i3}` | :math:`a_{i4}` | :math:`a_{i5}` | :math:`a_{i6}` | :math:`a_{i7}` |
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+===============================+===============================+===============================+===============================+===============================+===============================+===============================+===============================+
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| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
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+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+
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| 0.276393202250021 | 0.276393202250021 | 0 | 0 | 0 | 0 | 0 | 0 |
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+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+
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| 0.723606797749979 | -0.223606797749979 | 0.947213595499958 | 0 | 0 | 0 | 0 | 0 |
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+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+
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| 0.276393202250021 | 0.0326237921249264 | 0.309016994374947 | -0.0652475842498529 | 0 | 0 | 0 | 0 |
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+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+
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| 0.723606797749979 | 0.0460655337083368 | 0 | 0.166666666666667 | 0.510874597374975 | 0 | 0 | 0 |
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+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+
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| 0.276393202250021 | 0.12060113295833 | 0 | -0.181694990624912 | 0.166666666666667 | 0.170820393249937 | 0 | 0 |
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+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+
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| 1 | 0.166666666666667 | 0 | 0.0751416197912285 | -3.38770632020821 | 0.52786404500042 | 3.61803398874989 | 0 |
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+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+
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| b_i | 0.0833333333333333 | 0 | 0 | 0 | 0.416666666666667 | 0.416666666666667 | 0.0833333333333333 |
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+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+
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‎_sources/9_references.rst.txt‎

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.. [Lekien2005] Lekien, F., Marsden, J. "Tricubic interpolation in three dimensions," *International Journal for Numerical Methods in Engineering* **63** (2005).
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.. [Butcher] Butcher J. C. "On Runge-Kutta processes of high order." *Journal of the Australian Mathematical Society* **4(2)** (1964). doi:10.1017/S1446788700023387
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.. [Butcher1964] Butcher J. C. "On Runge-Kutta processes of high order." *Journal of the Australian Mathematical Society* **4(2)** (1964). doi:10.1017/S1446788700023387
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Papers Who Uses ``Py3DFTLE``

‎objects.inv‎

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‎searchindex.js‎

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