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Update 3_numerical.rst
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‎source/3_numerical.rst‎

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@@ -90,44 +90,26 @@ 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:`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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+----------+-----------------------------+-------------------------------+----------------------------+---------------------------------+------------------------------+-----------------------------+--------------------------+
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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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