How to get the Radau 5th order equations give the Butcher tableau 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 :)
 A: 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?
