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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

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

numaaa
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1 Answers1

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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.

Thomas
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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 04 '13 at 08:51
  • 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: http://en.wikipedia.org/wiki/Runge%E2%80%93Kutta_methods#Explicit_Runge.E2.80.93Kutta_methods – Thomas Feb 04 '13 at 08:58
  • Im trying to understand why we must set c1=0, in rk methods. In the context above, what i mean with c1 is the first element on the left column of butcher tableau. What i found is that we set it equal to zero so the method becomes explicit, which doent make sense when i write down the equations. The method is explicit even when c1 isnt zero. Any tips on litterature that explains this? – JustANoob Nov 13 '18 at 10:24