Alternating-Direction Line-Relaxation Methods on Multicomputers

Abstract

Abstract. We study the multicomputer performance of a three-dimensional Navier-Stokes solver based on alternating-direction line-relaxation methods. We compare several multicomputer implementations, each of which combines a particular line-relaxation method and a particular distributed block-tridiagonal solver. In our experiments, the problem size was determined by resolution requirements of the application. As a result, the granularity of the computations of our study is finer than is customary in the performance analysis of concurrent block-tridiagonal solvers. Our best results were obtained with a modified half-Gauss-Seidel line-relaxation method implemented by means of a new iterative block-tridiagonal solver that is developed here. Most computations were performed on the Intel Touchstone Delta, but we also used the Intel Paragon XP]S, the Parsytec SC-256, and the Fujitsu S-600 for comparison.Key words. Navier-Stokes equations, concurrency, parallelism, block-tridiagonal systems, tridiagonal systems, ADI, alternating directions AMS subject classifications. 65M20, 65N40, 65Y05, 76C05, 76M201. Introduction. When using alternating-direction line-relaxation methods for systems of partial-differential equations discretized on a rectangular grid, one must solve many blocktridiagonal systems of linear equations in every relaxation step. This type of computation surfaces in many applications. In our work, we faced it when parallelizing a highly vectorized solver for the three-dimensional, unsteady, and incompressible Navier-Stokes equations [4] for use on multicomputers. In this paper, we shall discuss and analyze five line-relaxation methods and six distributed block-tridiagonal solvers used during the course of this project. We have measured the performance of several combinations of relaxation methods and block-

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