On the efficiency of a SOR-like method suited to vector processors

Abstract

A simple implementation on vector processors of the SOR method for solving a linear system of equations arising from the discretization of partial differential equations changes the SOR method into another, which, though, looks like the SOR method. In this paper, the efficiency of this SOR-like method (pseudo-SOR method) is investigated. For the Poisson equation in a rectangular region, in both five-point and nine-point discretization, we prove analytically that the optimal acceleration parameter is smaller and the optimal convergence rate is lower in the pseudo-SOR method. Comparison on several vector computers was made between the SOR method vectorized with the hyperplane technique and the pseudo-SOR method. It turned out that the hyperplane SOR is superior to the pseudo-SOR method although the latter is easily vectorizable on vector processors. We also tested an example of the Poisson equation discretized in curved coordinates in a region bounded by two eccentric circles and found numerically that the optimal acceleration parameter becomes small and the convergence becomes slow in the pseudo-SOR method, while the optimal acceleration parameter in the SOR method does not change appreciably. The genuine SOR method is superior to the pseudo-SOR method.