PARALLELIZATION OF A CLASS OF IMPLICIT FINITE DIFFERENCE SCHEMES IN COMPUTATIONAL FLUID DYNAMICS

Abstract

Implicit finite difference schemes are often the preferred numerical methods in computational fluid dynamics, requiring less stringent stability bounds than the explicit schemes. Each iteration in an implicit scheme, however, involves global data dependencies in the form of second and higher order recurrences. Efficient parallel implementations of such methods, therefore, are more difficult and can be non-intuitive. Moreover, in practical applications, some sections of the computations are explicit while other sections are implicit in nature. Developing techniques that extract maximum parallelism simultaneously from all sections of an application is considerably harder than extracting parallelism from just an implicit section or an explicit section of the computations. In this paper, we consider issues in the parallelization of techniques that are used for solving Euler and thin-layer Navier-Stokes equations and that require, as a part of the computation, solving large linear systems in the form of block tri-diagonal and/or scalar penta-diagonal matrices. We focus our attention on three-dimensional cases and present schemes that minimize the total execution time when implemented on a message passing system. We describe various partitioning and scheduling strategies for alleviating the effects of the global data dependencies. Analyses of the computation, the communication, and the memory requirements of these methods are presented. The ARC-3D code, developed at NASA Ames, is used as an example application. Performance of the proposed methods is verified on the Victor multiprocessor system, a message passing architecture developed at the IBM T.J. Watson Research Center.