Hybrid Methods for Initial Value Problems in Ordinary Differential ...

HYBRID METHODS FOR INITIAL VALUE PROBLEMS
hold for k = 0 when the equations represent the second order Runge-Kutta
method. The question of whether this is true for all k naturally arises. Un-
fortunately the answer is no, but it is true for k = 3 and 4, giving rise to
8th and 10th order methods respectively. Parameters for both cases are
given below, but the 10th order method has undesirable stability proper-
ties due to large roots of the stability polynomial and hence it is not recom-
mended. For k = 5, 6, 7, 8, and 9, methods of order 2k + 2 are not stable.
These results were obtained by a numerical investigation of methods
for k = 3, 4, -.. , 9 on ILLIAC 11. For each k, the coefficients of the
method were calculated from the 2k + 3 linear simultaneous equations
obtained by a Taylor series expansion to 2k + 3 terms. Since there are
2k + 4 parameters in the corrector, these coefficients are linear functions
of one parameter, which was taken as yzl, the coefficient of f,+l. The roots
of the stability polynomial,
ffj0
Pjo
ffjl
Pjl
ffj2
Pj2
ffj3
014
Y$0
3jl
1
TABLE5
Stable methods for k = 3 and 4
Half point ( j = 0) Predictor j = 1 I
Corrector (j = 2)
k = 3
8th Order
83
k = 4
10th Order
(Rounded to
9 digits)