# butcher tableau for given algorithm

Given $y'(t) = f(t,y(t))$ and the following algorithm:

$$y_{n+\frac{1}{2}} = y_n + \frac{h}{2}f(t_n,y_n)$$ $$y_{n+1} = y_n + hf(t_n+\frac{h}{2},y_{n+\frac{1}{2}})$$

We should show that this can be seen as an explicit runge kutta algorithm (how to exactly do this?) and determine the butcher tableu.

I somehow compared this to the usual form in which an runge kutte algorithm is given and think the tableu must be

$$\begin{array}{c|cc} 0 & 0 & 0 \\ \frac{1}{2} & \frac{1}{2} & 0 \\ \hline & 0 & 1 \\ \end{array}$$

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Your tableu is correct. Your method is equivalent to the one mentioned here. A Runge-Kutta method is explicit, if the corresponding Butcher tableau has all diagonal and upper diagonal elements equal to zero. This is the case in your example.

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Thanks. Is there a clear way of showing that the method can be seen as a Runge-Kutta method? I just compared it kind of symbolically and rearranged somehow the formula.. –  numaaa Feb 4 '13 at 8:51
Try to reformulate your two equation by substituting $y_{n+1/2}$ in the second equation with the first equation to obtain an expression of the form, given in the reference: en.wikipedia.org/wiki/… –  sonystarmap Feb 4 '13 at 8:58