Solving ordinary differential equations on parallel computers — applied to dynamic rolling bearings simulation

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Abstract

In this paper we investigate how to solve certain kinds of ordinary differential equations (ODEs) efficiently on several types of MIMD parallel computers. The amount of parallelism for solving initial value problems such as ODEs is often quite limited, but by exploiting some characteristics of the application area where these problems are solved, the amount of parallelism can be increased. We focus on solving ODEs for rolling bearing dynamics simulation, which is computationally expensive. Typical characteristics of such ODEs are: stiff ODEs, very high numerical precision needed for solution, and computationally expensive to evaluate the derivatives.

We have adapted conventional solvers such as LSODA for execution on parallel computers, for example by evaluating the right-hand sides of the ODEs in parallel. The parallel machines used are: a Parsytec GigaCube with 16 T805 processors using the PARIX operating system, a Sun SPARCcenter 2000 with 8 processors and Solaris 2.3, and a cluster of nine SPARC 10 workstations connected via ethernet and using PVM. All these can be considered as Multiple Instruction Multiple Data (MIMD) architectures.

The obtained speedup was fairly good, approximately two thirds of linear speedup. However, this application requires rather fine-grained synchronization, which favours scheduling methods that minimize communication. As always, it is easier to get good speedups on machines with slower processors.