Efficient eighth order p-stable methods for second order initial value problems

Abstract

A family of eighth order P-stable methods for solving second order initial value problems is considered. The nonlinear algebraic systems, which results on applying one of the methods in this family to a nonlinear differential system, may be solved by using a modified Newton method. We derive methods which require only four (new) function evaluations per iteration and for which the iteration matrix is a true real perfect quartic. This means that at most one real matrix must be factorised at each step. Finally numerical results are presented to illustrate our local error estimation technique.