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 as1 … as,s-1 b1 b2 … bs-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 as1 … as,s-1 b1 b2 … bs-1 bs b*1 b*2 … b*s-1 b*s