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====== Butcher tableau ======

The Butcher tableau after John C. Butcher is an arrangement of coefficients to be applied in Runge-Kutta methods.

A generalization of a Runge-Kutta method is given by:

//y<sub>n+1</sub> = y<sub>n</sub> + hb<sub>i</sub>k<sub>i</sub>, i=1:s//

where

//k<sub>1</sub> = f(t<sub>n</sub>,y<sub>n</sub>),//

//k<sub>2</sub> = f(t<sub>n</sub>+c<sub>2</sub>h,y<sub>n</sub>+a<sub>21</sub>hk<sub>1</sub>),//

//k<sub>3</sub> = f(t<sub>n</sub>+c<sub>3</sub>h,y<sub>n</sub>+a<sub>31</sub>hk<sub>1</sub>+a<sub>32</sub>hk<sub>2</sub>),//

…

//k<sub>s</sub> = f(t<sub>n</sub>+c<sub>s</sub>h,y<sub>n</sub>+a<sub>s1</sub>hk<sub>1</sub>+a<sub>s2</sub>hk<sub>2</sub>+…+a<sub>s,s-1</sub>hk<sub>s-1</sub>).//

Corresponding Butcher tableau is:

0
//c<sub>2</sub>//
//a<sub>21</sub>//
//c<sub>3</sub>//
//a<sub>31</sub>//
//a<sub>32</sub>//
...
...
...
//c<sub>s</sub>//
//a<sub>s1</sub>//
//a<sub>s1</sub>//
...
//a<sub>s,s-1</sub>//
//b<sub>1</sub>//
//b<sub>2</sub>//
...
//b<sub>s-1</sub>//
//b<sub>s</sub>//
For adaptive Runge-Kutta methods the local truncation error is estimated by the usage of two methods in the tableau, one with order //p// and one with order //p//-1.

The lower order step is given by

//y<sup>*</sup><sub>n+1</sub> = y<sub>n</sub> + hb<sup>*</sup><sub>i</sub>k<sub>i</sub>, i=1:s//

where the_k<sub>i</sub>_ are the same as for the higher order method. Then the error is

//e<sub>n+1</sub> = y<sub>n+1</sub>-y<sup>*</sup><sub>n+1</sub> = h(b<sub>i</sub>-b<sup>*</sup><sub>i</sub>)k<sub>i</sub>, i=1:s//,

which is //O(h<sup>p</sup>)//. The Butcher tableau for adaptive RK methods is extended to give the values of **//b<sup>*</sup><sub>i</sub>//**:

0
//c<sub>2</sub>//
//a<sub>21</sub>//
//c<sub>3</sub>//
//a<sub>31</sub>//
//a<sub>32</sub>//
...
...
...
//c<sub>s</sub>//
//a<sub>s1</sub>//
//a<sub>s1</sub>//
...
//a<sub>s,s-1</sub>//
//b<sub>1</sub>//
//b<sub>2</sub>//
...
//b<sub>s-1</sub>//
//b<sub>s</sub>//
**//b<sup>*</sup><sub>1</sub>//**
**//b<sup>*</sup><sub>2</sub>//**
**...**
**//b<sup>*</sup><sub>s-1</sub>//**
**//b<sup>*</sup><sub>s</sub>//**
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