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‎.doctrees/3_numerical.doctree‎

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‎.doctrees/environment.pickle‎

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‎3_numerical.html‎

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\mathbf{x}_{n+1} = \mathbf{x}_n + \Delta t\bigl(\tfrac{1}{20}k_1 + \tfrac{1}{4}k_4 + \tfrac{1}{5}k_5 + \tfrac{1}{2}k_6\bigr).\end{split}\]</div>
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<p>This scheme incurs a global error of order with six velocity evaluations.</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-odd"><td><p><span class="math notranslate nohighlight">\(0\)</span></p></td>
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<tr class="row-even"><td><p><span class="math notranslate nohighlight">\(0\)</span></p></td>
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<td></td>
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<td></td>
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<td></td>
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<td></td>
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<td></td>
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<td></td>
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<td></td>
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</tr>
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<tr class="row-even"><td><p><span class="math notranslate nohighlight">\((5\mp\sqrt{5})/10\)</span></p></td>
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<tr class="row-odd"><td><p><span class="math notranslate nohighlight">\((5\mp\sqrt{5})/10\)</span></p></td>
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<td><p><span class="math notranslate nohighlight">\((5\mp\sqrt{5})/10\)</span></p></td>
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<td></td>
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<td></td>
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<td></td>
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<td></td>
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<td></td>
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<td></td>
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</tr>
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<tr class="row-odd"><td><p><span class="math notranslate nohighlight">\((5\pm\sqrt{5})/10\)</span></p></td>
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<tr class="row-even"><td><p><span class="math notranslate nohighlight">\((5\pm\sqrt{5})/10\)</span></p></td>
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<td><p><span class="math notranslate nohighlight">\(\mp\sqrt{5}/10\)</span></p></td>
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<td><p><span class="math notranslate nohighlight">\((5\pm2\sqrt{5})/10\)</span></p></td>
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<td></td>
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<td></td>
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<td></td>
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<td></td>
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<td></td>
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</tr>
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<tr class="row-even"><td><p><span class="math notranslate nohighlight">\((5\mp\sqrt{5})/10\)</span></p></td>
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<tr class="row-odd"><td><p><span class="math notranslate nohighlight">\((5\mp\sqrt{5})/10\)</span></p></td>
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<td><p><span class="math notranslate nohighlight">\((-15\pm7\sqrt{5})/20\)</span></p></td>
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<td><p><span class="math notranslate nohighlight">\((-1\pm\sqrt{5})/4\)</span></p></td>
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<td><p><span class="math notranslate nohighlight">\((15\mp7\sqrt{5})/10\)</span></p></td>
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<td></td>
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<td></td>
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<td></td>
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<td></td>
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</tr>
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<tr class="row-odd"><td><p><span class="math notranslate nohighlight">\((5\pm\sqrt{5})/10\)</span></p></td>
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<tr class="row-even"><td><p><span class="math notranslate nohighlight">\((5\pm\sqrt{5})/10\)</span></p></td>
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<td><p><span class="math notranslate nohighlight">\((5\mp\sqrt{5})/60\)</span></p></td>
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<td><p><span class="math notranslate nohighlight">\(0\)</span></p></td>
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<td><p><span class="math notranslate nohighlight">\(1/6\)</span></p></td>
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<td><p><span class="math notranslate nohighlight">\((15\pm7\sqrt{5})/60\)</span></p></td>
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<td></td>
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<td></td>
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<td></td>
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</tr>
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<tr class="row-even"><td><p><span class="math notranslate nohighlight">\((5\mp\sqrt{5})/10\)</span></p></td>
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<tr class="row-odd"><td><p><span class="math notranslate nohighlight">\((5\mp\sqrt{5})/10\)</span></p></td>
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<td><p><span class="math notranslate nohighlight">\((5\pm\sqrt{5})/60\)</span></p></td>
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<td><p><span class="math notranslate nohighlight">\(0\)</span></p></td>
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<td><p><span class="math notranslate nohighlight">\((9\mp5\sqrt{5})/12\)</span></p></td>
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<td><p><span class="math notranslate nohighlight">\(1/6\)</span></p></td>
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<td><p><span class="math notranslate nohighlight">\((-5\pm3\sqrt{5})/10\)</span></p></td>
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<td></td>
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<td></td>
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</tr>
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<tr class="row-odd"><td><p><span class="math notranslate nohighlight">\(1\)</span></p></td>
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<tr class="row-even"><td><p><span class="math notranslate nohighlight">\(1\)</span></p></td>
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<td><p><span class="math notranslate nohighlight">\(1/6\)</span></p></td>
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<td><p><span class="math notranslate nohighlight">\(0\)</span></p></td>
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<td><p><span class="math notranslate nohighlight">\((-55\pm25\sqrt{5})/12\)</span></p></td>
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<td><p><span class="math notranslate nohighlight">\((-25\mp7\sqrt{5})/12\)</span></p></td>
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<td><p><span class="math notranslate nohighlight">\(5\mp2\sqrt{5}\)</span></p></td>
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<td><p><span class="math notranslate nohighlight">\((5\pm\sqrt{5})/2\)</span></p></td>
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<td></td>
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</tr>
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<tr class="row-even"><td><p><span class="math notranslate nohighlight">\(1/12\)</span></p></td>
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<tr class="row-odd"><td><p><span class="math notranslate nohighlight">\(b_i\)</span></p></td>
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<td><p><span class="math notranslate nohighlight">\(1/12\)</span></p></td>
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<td><p><span class="math notranslate nohighlight">\(0\)</span></p></td>
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<td><p><span class="math notranslate nohighlight">\(0\)</span></p></td>
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<td><p><span class="math notranslate nohighlight">\(0\)</span></p></td>

‎_sources/3_numerical.rst.txt‎

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@@ -89,45 +89,28 @@ The seven-stage scheme uses non-uniform weights to attain global :math:`O(\Delta
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This scheme incurs a global error of order with six velocity evaluations.
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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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| :math:`0` | | | | | | | |
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+-----------------------------+-------------------------------+----------------------------+---------------------------------+------------------------------+-----------------------------+--------------------------+-------------------------+
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| :math:`(5\mp\sqrt{5})/10` | :math:`(5\mp\sqrt{5})/10` | | | | | | |
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+-----------------------------+-------------------------------+----------------------------+---------------------------------+------------------------------+-----------------------------+--------------------------+-------------------------+
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| :math:`(5\pm\sqrt{5})/10` | :math:`\mp\sqrt{5}/10` | :math:`(5\pm2\sqrt{5})/10` | | | | | |
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+-----------------------------+-------------------------------+----------------------------+---------------------------------+------------------------------+-----------------------------+--------------------------+-------------------------+
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| :math:`(5\mp\sqrt{5})/10` | :math:`(-15\pm7\sqrt{5})/20` | :math:`(-1\pm\sqrt{5})/4` | :math:`(15\mp7\sqrt{5})/10` | | | | |
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+-----------------------------+-------------------------------+----------------------------+---------------------------------+------------------------------+-----------------------------+--------------------------+-------------------------+
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| :math:`(5\pm\sqrt{5})/10` | :math:`(5\mp\sqrt{5})/60` | :math:`0` | :math:`1/6` | :math:`(15\pm7\sqrt{5})/60` | | | |
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+-----------------------------+-------------------------------+----------------------------+---------------------------------+------------------------------+-----------------------------+--------------------------+-------------------------+
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| :math:`(5\mp\sqrt{5})/10` | :math:`(5\pm\sqrt{5})/60` | :math:`0` | :math:`(9\mp5\sqrt{5})/12` | :math:`1/6` | :math:`(-5\pm3\sqrt{5})/10` | | |
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+-----------------------------+-------------------------------+----------------------------+---------------------------------+------------------------------+-----------------------------+--------------------------+-------------------------+
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| :math:`1` | :math:`1/6` | :math:`0` | :math:`(-55\pm25\sqrt{5})/12` | :math:`(-25\mp7\sqrt{5})/12` | :math:`5\mp2\sqrt{5}` | :math:`(5\pm\sqrt{5})/2` | |
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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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+-----------------------------+-------------------------------+----------------------------+---------------------------------+------------------------------+-----------------------------+--------------------------+
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| :math:`0` | | | | | | |
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+-----------------------------+-------------------------------+----------------------------+---------------------------------+------------------------------+-----------------------------+--------------------------+
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| :math:`(5\mp\sqrt{5})/10` | :math:`(5\mp\sqrt{5})/10` | | | | | |
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+-----------------------------+-------------------------------+----------------------------+---------------------------------+------------------------------+-----------------------------+--------------------------+
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| :math:`(5\pm\sqrt{5})/10` | :math:`\mp\sqrt{5}/10` | :math:`(5\pm2\sqrt{5})/10` | | | | |
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+-----------------------------+-------------------------------+----------------------------+---------------------------------+------------------------------+-----------------------------+--------------------------+
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| :math:`(5\mp\sqrt{5})/10` | :math:`(-15\pm7\sqrt{5})/20` | :math:`(-1\pm\sqrt{5})/4` | :math:`(15\mp7\sqrt{5})/10` | | | |
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+-----------------------------+-------------------------------+----------------------------+---------------------------------+------------------------------+-----------------------------+--------------------------+
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| :math:`(5\pm\sqrt{5})/10` | :math:`(5\mp\sqrt{5})/60` | :math:`0` | :math:`1/6` | :math:`(15\pm7\sqrt{5})/60` | | |
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+-----------------------------+-------------------------------+----------------------------+---------------------------------+------------------------------+-----------------------------+--------------------------+
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| :math:`(5\mp\sqrt{5})/10` | :math:`(5\pm\sqrt{5})/60` | :math:`0` | :math:`(9\mp5\sqrt{5})/12` | :math:`1/6` | :math:`(-5\pm3\sqrt{5})/10` | |
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+-----------------------------+-------------------------------+----------------------------+---------------------------------+------------------------------+-----------------------------+--------------------------+
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| :math:`1` | :math:`1/6` | :math:`0` | :math:`(-55\pm25\sqrt{5})/12` | :math:`(-25\mp7\sqrt{5})/12` | :math:`5\mp2\sqrt{5}` | :math:`(5\pm\sqrt{5})/2` |
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+-----------------------------+-------------------------------+----------------------------+---------------------------------+------------------------------+-----------------------------+--------------------------+
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| :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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+-----------------------------------+-+-------------------------------------+----------------------------------+---------------------------------------+------------------------------------+-----------------------------------+--------------------------------+
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| :math:`0` |丨| | | | | | |
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+-----------------------------------+-+-------------------------------------+----------------------------------+---------------------------------------+------------------------------------+-----------------------------------+--------------------------------+
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| :math:`\frac{5\mp\sqrt{5}}{10}` |丨| :math:`\frac{5\mp\sqrt{5}}{10}` | | | | | |
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+-----------------------------------+-+-------------------------------------+----------------------------------+---------------------------------------+------------------------------------+-----------------------------------+--------------------------------+
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| :math:`\frac{5\pm\sqrt{5}}{10}` |丨| :math:`\frac{\mp\sqrt{5}}{10}` | :math:`\frac{5\pm2\sqrt{5}}{10}` | | | | |
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+-----------------------------------+-+-------------------------------------+----------------------------------+---------------------------------------+------------------------------------+-----------------------------------+--------------------------------+
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| :math:`\frac{5\mp\sqrt{5}}{10}` |丨| :math:`\frac{-15\pm7\sqrt{5}}{20}` | :math:`\frac{-1\pm\sqrt{5}}{4}` | :math:`\frac{15\mp7\sqrt{5}}{10}` | | | |
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+-----------------------------------+-+-------------------------------------+----------------------------------+---------------------------------------+------------------------------------+-----------------------------------+--------------------------------+
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| :math:`\frac{5\pm\sqrt{5}}{10}` |丨| :math:`\frac{5\mp\sqrt{5}}{60}` | :math:`0` | :math:`\frac{1}{6}` | :math:`\frac{15\pm7\sqrt{5}}{60}` | | |
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+-----------------------------------+-+-------------------------------------+----------------------------------+---------------------------------------+------------------------------------+-----------------------------------+--------------------------------+
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| :math:`\frac{5\mp\sqrt{5}}{10}` |丨| :math:`\frac{5\pm\sqrt{5}}{60}` | :math:`0` | :math:`\frac{9\mp5\sqrt{5}}{12}` | :math:`\frac{1}{6}` | :math:`\frac{-5\pm3\sqrt{5}}{10}` | |
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+-----------------------------------+-+-------------------------------------+----------------------------------+---------------------------------------+------------------------------------+-----------------------------------+--------------------------------+
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| :math:`1` |丨| :math:`\frac{1}{6}` | :math:`0` | :math:`\frac{-55\pm25\sqrt{5}}{12}` | :math:`\frac{-25\mp7\sqrt{5}}{12}` | :math:`5\mp2\sqrt{5}` | :math:`\frac{5\pm\sqrt{5}}{2}` |
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+-----------------------------------+-+-------------------------------------+----------------------------------+---------------------------------------+------------------------------------+-----------------------------------+--------------------------------+
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| ———————— |丨| ———————— | ———————— | ———————— | ———————— | ———————— | ———————— |
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+-----------------------------------+-+-------------------------------------+----------------------------------+---------------------------------------+------------------------------------+-----------------------------------+--------------------------------+
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| :math:`\frac{1}{12}` |丨| :math:`0` | :math:`0` | :math:`0` | :math:`\frac{5}{12}` | :math:`\frac{5}{12}` | :math:`\frac{1}{12}` |
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+-----------------------------------+-+-------------------------------------+----------------------------------+---------------------------------------+------------------------------------+-----------------------------------+--------------------------------+
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