High orderP-stable formulae for the numerical integration of periodic initial value problems

Abstract

Recently there has been considerable interest in the approximate numerical integration of the special initial value problemy?=f(x, y) for cases where it is known in advance that the required solution is periodic. The well known class of Stormer-Cowell methods with stepnumber greater than 2 exhibit orbital instability and so are often unsuitable for the integration of such problems. An appropriate stability requirement for the numerical integration of periodic problems is that ofP-stability. However Lambert and Watson have shown that aP-stable linear multistep method cannot have an order of accuracy greater than 2. In the present paper a class of 2-step methods of Runge-Kutta type is discussed for the numerical solution of periodic initial value problems.P-stable formulae with orders up to 6 are derived and these are shown to compare favourably with existing methods.