Parallel iterated Runge-Kutta methods and applications

Abstract

The iterated Runge-Kutta (IRK) method is an iteration scheme for the numerical solution of initial value problems (IVP) of ordinary differential equations (ODEs) that is based on a predictor-corrector method with an Runge-Kutta (RK) method as corrector. Embedded approxination formulae are used to control stepsize. We present different parallel algorithms of the IRK method on distributed memory multiprocessors for the solution of systems of ODEs. The parallel algorithms are given in an SPMD (single-program multipledata) programming style where data exchanges are described with appropriate communication primitives. A theoretical performance analysis and a runtime simulation allow to value the presented algorithms. The implementation on the Intel iPSC/860 confirms the predicted runtimes. The speedup values strongly depend on the particular system of ODEs to be solved. The parallel IRK method is applied to a typical discretization problem, the discretized Brusselator equation. Application specific modifications of the general parallel ODE solver are developped which result in a considerable reduction of the parallel execution time.,