Update 3_numerical.rst

Author: hweifluids

source/3_numerical.rst

@@ -74,7 +74,8 @@ This yields a global error of order :math:`O(\Delta t^4)` with four velocity eva
 
 **Sixth-Order Runge-Kutta (RK6)**
 
-The seven-stage scheme uses non-uniform weights to attain global :math:`O(\Delta t^6)` accuracy. This method originates from 
+The seven-stage scheme uses non-uniform weights to attain global :math:`O(\Delta t^6)` accuracy. This method originates from [Butcher]_, as well as the coefficients (Butcher table) listed in the basically-tailest table of the reference.
+
 
 .. math::
 
@@ -108,6 +109,25 @@ This scheme incurs a global error of order  with six velocity evaluations.
 +-----------------------------+-------------------------------+----------------------------+---------------------------------+------------------------------+-----------------------------+--------------------------+
 
 
++-----------------------------------+-------------------------------------+----------------------------------+---------------------------------------+------------------------------------+-----------------------------------+--------------------------------+
+| :math:`0`                         |                                     |                                  |                                       |                                    |                                   |                                |
++-----------------------------------+-------------------------------------+----------------------------------+---------------------------------------+------------------------------------+-----------------------------------+--------------------------------+
+| :math:`\frac{5\mp\sqrt{5}}{10}`   | :math:`\frac{5\mp\sqrt{5}}{10}`     |                                  |                                       |                                    |                                   |                                |
++-----------------------------------+-------------------------------------+----------------------------------+---------------------------------------+------------------------------------+-----------------------------------+--------------------------------+
+| :math:`\frac{5\pm\sqrt{5}}{10}`   | :math:`\frac{\mp\sqrt{5}}{10}`      | :math:`\frac{5\pm2\sqrt{5}}{10}` |                                       |                                    |                                   |                                |
++-----------------------------------+-------------------------------------+----------------------------------+---------------------------------------+------------------------------------+-----------------------------------+--------------------------------+
+| :math:`\frac{5\mp\sqrt{5}}{10}`   | :math:`\frac{-15\pm7\sqrt{5}}{20}`  | :math:`\frac{-1\pm\sqrt{5}}{4}`  | :math:`\frac{15\mp7\sqrt{5}}{10}`     |                                    |                                   |                                |
++-----------------------------------+-------------------------------------+----------------------------------+---------------------------------------+------------------------------------+-----------------------------------+--------------------------------+
+| :math:`\frac{5\pm\sqrt{5}}{10}`   | :math:`\frac{5\mp\sqrt{5}}{60}`     | :math:`0`                        | :math:`\frac{1}{6}`                   | :math:`\frac{15\pm7\sqrt{5}}{60}`  |                                   |                                |
++-----------------------------------+-------------------------------------+----------------------------------+---------------------------------------+------------------------------------+-----------------------------------+--------------------------------+
+| :math:`\frac{5\mp\sqrt{5}}{10}`   | :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}` |                                |
++-----------------------------------+-------------------------------------+----------------------------------+---------------------------------------+------------------------------------+-----------------------------------+--------------------------------+
+| :math:`1`                         | :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}` |
++-----------------------------------+-------------------------------------+----------------------------------+---------------------------------------+------------------------------------+-----------------------------------+--------------------------------+
+| :math:`\frac{1}{12}`              | :math:`0`                           | :math:`0`                        | :math:`0`                             | :math:`\frac{5}{12}`               | :math:`\frac{5}{12}`              | :math:`\frac{1}{12}`           |
++-----------------------------------+-------------------------------------+----------------------------------+---------------------------------------+------------------------------------+-----------------------------------+--------------------------------+
+
+
 
 .. _ftlefinal: