Parallel Implementations of Iterated Runge-Kutta Methods

Abstract

We investigate different parallel algorithms for the iter ated Runge-Kutta method on distributed memory mul tiprocessors for the solution of systems of ordinary differential equations (ODEs). The iterated Runge-Kutta method is an iteration scheme for the numerical solu tion of initial-value problems of nonstiff ODEs; embed ded approximation formulae are used to control the step size. The parallel algorithms realize potential par allelism across the method and the system in a group- SPMD (single-program, multiple-data) programming style, using an appropriate set of communication prim itives that can be implemented on all common topolo gies. A theoretical performance analysis with run-time formulae and a run-time simulation show the value of the algorithms. The implementation on the Intel iPSC/ 860 confirms the predicted run times. The speedup values depend strongly on the particular system of ODEs to be solved. The parallel iterated Runge-Kutta method is applied to a typical discretization problem, the discretized Brusselator equation. Application-spe cific modifications of the general parallel ODE solver are developed, which result in a considerable reduc tion in the parallel execution time.