Stability of Parallel Explicit ODE Methods

Abstract

General-purpose numerical methods for ordinary initial value problems are usually found in the class of linear multistage multivalue methods, first formulated by J. C. Butcher. Among these the explicit methods are easiest to implement. For this reason there has been considerable research activity devoted to generating methods of this class which utilize independent function evaluations that can be performed in par- allel. Each such group of concurrent function evaluations can be regarded as a stage of the method. It is desirable that methods have a large region of stability so that stability-dictated restrictions on the stepsize will occur less frequently. In order to do a fair comparison of stability regions, one should consider the scaled stability region. This is defined using not the nominal stepsize but rather the stepsize per unit of work. Each stage counts as one work unit; in the context of parallel computing each stage may involve any number of independent function evaluations. Very important results on scaled stability regions were obtained by Jeltsch and Nevanlinna, in the context of serial computation. They identify a type of method, actually a type of alge- braic function, whose scaled stability region is optimal. This means that the region is not a proper subset of some other scaled stability region. The distinguished type of algebraic functions is one that satisfies Property C, which requires that the principal branch of the algebraic function be analytic outside the stability region except for a pole at infinity whose multiplicity equals the number of work units of the method. It is shown that by modifying Property C the result on optimal stability regions holds also in the context of parallel computa- tion. An important consequence is that, although the use of parallelism enlarges the class of scaled stability regions, it does not result in scaled stability regions that properly contain those obtained without parallelism. Roughly stated, parallelism cannot improve the stability of an explicit linear method; but it can, even for the standard test problem, improve the accuracy.