How to get the Radau 5th order equations give the Butcher tableau

Question

I'm don't well understand how from the tableau of Butcher I can find the equations needed for implement the method, In particular I want be able to know the equations for Radau method 4th and 5th order ! here is reported the Radau Table .. may somebody help me to discover the equations to implement ?? thanks :)

Answer 1

It works similar to your other question here.

Let $y'(t) = f(t,y(t))$, $y(t_0)=y_0$ be your initial value problem. Set $u_0 = y_0$. Then the $i+1$-th iteration of a runge-kutta method with $s$ stages is defined as

$$ u_{i+1} = u_i + h \sum_{j=1}^{s} b_j \cdot f(t_i+c_j \cdot h,\, u^{(j)}_{i+1}) \\ u^{(j)}_{i+1} = u_i + h \sum_{k=1}^{s} a_{jk} \cdot f(t_i + c_k \cdot h,\, u^{(k)}_{i+1}) $$

where

$$\begin{array}{c|ccc} c_1 & a_{11} & \cdots & a_{1s}\\ \vdots & \vdots & \ddots & \vdots \\ c_s & a_{s1} & \cdots & a_{ss} \\ \hline & b_1 & \cdots & b_s \end{array} $$

is the given butcher tableau. For the fifth-order Radau IIA:

You just have to insert the $b_1,\ldots,b_s$ and $c_1,\ldots,c_s$ into the first equation and the $a_{jk}$ into the second equation. Using $$ \begin{align} b_1 &= \tfrac49 - \tfrac{\sqrt{6}}{36}, \\ b_2 &= \tfrac49 + \tfrac{\sqrt{6}}{36}, \\ b_3 &= \tfrac19, \\ c_1 &= \tfrac25 - \tfrac{\sqrt{6}}{10}, \\ c_2 &= \tfrac25 + \tfrac{\sqrt{6}}{10}, \\ c_3 &= 1 \end{align} $$

yields:

$$ \begin{align} u_{i+1} &= u_i + h \cdot \left( b_1 \cdot f(t_i+ c_1 h,\, u^{(1)}_{i+1}) + b_2 \cdot f(t_i+c_2 h,\, u^{(2)}_{i+1}) + b_3 \cdot f(t_i+c_3 h,\, u^{(3)}_{i+1}) \right) \\ &= u_i + h \cdot \left( (\tfrac49 - \tfrac{\sqrt{6}}{36}) \cdot f(t_i+ (\tfrac25 - \tfrac{\sqrt{6}}{10}) h,\, u^{(1)}_{i+1}) + (\tfrac49 + \tfrac{\sqrt{6}}{36}) \cdot f(t_i+(\tfrac25 + \tfrac{\sqrt{6}}{10}) h,\, u^{(2)}_{i+1}) + \tfrac19 \cdot f(t_i+1 h,\, u^{(3)}_{i+1}) \right) \end{align} $$ with $$ \begin{align} u^{(1)}_{i+1} &= u_i + h \cdot \left( a_{11} \cdot f(t_i + c_1 h,\, u^{(1)}_{i+1}) + a_{12} \cdot f(t_i + c_2 h,\, u^{(2)}_{i+1}) + a_{13} \cdot f(t_i + c_3 h,\, u^{(3)}_{i+1}) \right) \\ u^{(2)}_{i+1} &= u_i + h \cdot \left( a_{21} \cdot f(t_i + c_1 h,\, u^{(1)}_{i+1}) + a_{22} \cdot f(t_i + c_2 h,\, u^{(2)}_{i+1}) + a_{23} \cdot f(t_i + c_3 h,\, u^{(3)}_{i+1}) \right) \\ u^{(3)}_{i+1} &= u_i + h \cdot \left( a_{31} \cdot f(t_i + c_1 h,\, u^{(1)}_{i+1}) + a_{32} \cdot f(t_i + c_2 h,\, u^{(2)}_{i+1}) + a_{33} \cdot f(t_i + c_3 h,\, u^{(3)}_{i+1}) \right) \end{align} $$

Now the only thing left to do is inserting the $a_{jk}$ and $c_1,\ldots,c_3$ from the given butcher tableau. You can read from the butcher tableau: $a_{11}=\tfrac{11}{45}-\tfrac{7\sqrt{6}}{360},\, a_{12}=\tfrac{37}{225}-\tfrac{169\sqrt{6}}{1800},\, a_{13}=-\tfrac{2}{225}+\tfrac{\sqrt{6}}{75},\, a_{21} = \tfrac{37}{225}+\tfrac{169\sqrt{6}}{1800},\, \ldots$ Can you go on from here?