Families of two-step fourth order P-stable methods for second order differential equations

Abstract

This paper deals with a class of symmetric (hybrid) two-step fourth order P-stable methods for the numerical solution of special second order initial value problems. Such methods were proposed independently by Cash and Chawla and normally require three function evaluations per step. The purpose of this paper is to point out that there are some values of the (free) parameters available in the methods proposed which can reduce this work; we study two classes of such methods. This first is the class of 'economical' methods (see Definition 3.1) which reduce this work to two function evaluations per step, and the second is the class of 'efficient' methods (see Definition 3.2) which reduce this work with respect to implementation for nonlinear problems. We report numerical experiments to illustrate the order, acuracy and implementational aspects of these two classes of methods.