SOME RUNGE-KUTTA FORMULA PAIRS

In ["The Numerical Analysis of Ordinary Differential Equations," John Wiley, New York, 1987, pp. 298-303], Butcher derives a family of nine-stage formula pairs of orders 5 and 6. If the fifth-order approximation is propagated, only eight function evaluations are needed in each successful step. Here, that design is modified so that the sixth-order approximation can be propagated using only eight function evaluations in each successful step. The approach derives formulas for computing coefficients of a pair of methods in terms of five arbitrary parameters. The derivation is emphasized in anticipation of the construction of higher-order methods in continuing research. Of two particular methods selected for consideration, the first illustrates that an attempt to select an optimal pair from among a number of relatively efficient pairs on the basis of characteristic values of each pair may be ineffective. For the second pair, which appears to be as efficient as other known pairs, the region of absolute stability and C1-interpolants of global orders 5 and 6 are included.