Parallel preconditioning for CFD problems on the CM-5

Abstract

This chapter demonstrates a new approach for preconditioning of very large, sparse, unsymmetric, linear systems that can be successfully applied to computational fluid dynamics (CFD) problems on parallel machines. An approximate inverse is explicitly computed to the original matrix. This new preconditioning matrix can be applied most efficiently for iterative methods on massively parallel machines, because the preconditioning phase involves only a matrix-vector multiplication. For structured grid CFD problems, this involves mainly dense matrix operations. The actual computation of the preconditioning matrix has natural parallelism. For a problem of size “n,” the preconditioning matrix can be computed by solving “n” independent small least squares problems. The algorithm and its implementation on the Connection Machine CM-5 are discussed in the chapter. Its efficiency is demonstrated in the chapter with two CFD problems. The chapter concludes that the new preconditioned algorithm improves convergence considerably for problems with large Courant–Friedrichs–Lewy condition (CFL) number.