PARALLEL IMPLEMENTATIONS OF ITERATED RUNGE-KUTTA METHODS

We investigate different parallel algorithms for the iterated Runge-Ku tta method on distributed memory multiprocessors for the solution of s ystems of ordinary differential equations (ODEs). The iterated Runge-K utta method is an iteration scheme for the numerical solution of initi al-value problems of nonstiff ODEs; embedded approximation formulae ar e used to control the step size. The parallel algorithms realize poten tial parallelism across the method and the system in a group-SPMD (sin gle-program, multiple-data) programming style, using an appropriate se t of communication primitives that can be implemented on all common to pologies. A theoretical performance analysis with run-time formulae an d a run-time simulation show the value of the algorithms. The implemen tation on the Intel iPSC/860 confirms the predicted run times. The spe edup values depend strongly on the particular system of ODEs to be sol ved. The parallel iterated Runge-Kutta method is applied to a typical discretization problem, the discretized Brusselator equation. Applicat ion-specific modifications of the general parallel ODE solver are deve loped, which result in a considerable reduction in the parallel execut ion time.