Storage space reduction for the solution of systems of ordinary differential equations by pipelining and overlapping of vectors

Abstract

Systems of ordinary differential equations (ODEs) arise from the mathematical modeling of time-dependent processes. Many sequential and parallel numerical methods have been proposed that can simulate processes described by ODE systems with known initial state. One disadvantage common to the proposed methods is the large amount of storage space required if the ODE systems consist of many equations. Not only do they have to keep the solution of the ODE system corresponding to the current time step in memory, but also several intermediate solutions or results of evaluations of the right hand side function of the ODE system. In this paper, we present an approach based on pipelining and overlapping of vectors which can reduce the storage space of typical ODE solution methods such as Runge-Kutta (RK) and extrapolation methods. We analyze and compare the scalability of different implementation variants of embedded and iterated RK methods on several modern parallel computer systems. Our experiments show that, due to an increased locality of memory references, our approach leads to a good scalability behavior even on large numbers of processors.