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Update 3_numerical.rst
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‎source/3_numerical.rst‎

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@@ -89,24 +89,25 @@ 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:`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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