Parallel Solution of ODEs

Abstract

This talk examines the factors that will limit the application of parallel methods to ordinary differential equations (ODEs). Two primary classes of methods have been discussed in the literature: parallelism across time, also called parallelism across the method, in which different processors are used to handle different points in time (or different parts of the method); and parallelism across space, also called equation segmentation or parallelism across the system, in which different groups of equations are integrated on different processors. The latter class of methods is suitable and often proposed for real time applications, while both are suitable for non-real time applications. However, in both classes, the factor that limits the speed of the method is the inherent communication in the problem that must be represented in the method. In parallelism across time, the natural flow of information in the increasing time of an initial value problem is a limiting factor. It will be shown that it does not appear possible to achieve a speed of better than logarithmic in the number of time points, and reasons will be given why even that may be extremely difficult. In parallelism across space, the natural flow of information from one part of space to another in the simulated system maps into flow of information between processors. We examine the amount of information that must be transmitted. In a real time application, the time required for the information to flow across several processors may be the limiting factor.

Work supported in part by US DOE under grant DOE DEFG02-87ER25026 and by NSF under grant DMS 87-03226