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butcher_tableau

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:

yn+1 = yn + hbiki, i=1:s

where

k1 = f(tn,yn),

k2 = f(tn+c2h,yn+a21hk1),

k3 = f(tn+c3h,yn+a31hk1+a32hk2),

ks = f(tn+csh,yn+as1hk1+as2hk2+…+as,s-1hks-1).

Corresponding Butcher tableau is:

0 c2 a21 c3 a31 a32 … … … cs as1 as1as,s-1 b1 b2bs-1 bs 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*n+1 = yn + hb*iki, i=1:s

where the_ki_ are the same as for the higher order method. Then the error is

en+1 = yn+1-y*n+1 = h(bi-b*i)ki, i=1:s,

which is O(hp). The Butcher tableau for adaptive RK methods is extended to give the values of b*i:

0 c2 a21 c3 a31 a32 … … … cs as1 as1as,s-1 b1 b2bs-1 bs b*1 b*2 b*s-1 b*s

butcher_tableau.txt · Last modified: 2023/08/09 12:05 by mina