Stationary, nonstationary, and hybrid iterative method of moments solution schemes

Abstract

The purpose of this paper is to report on the convergence rates of two iterative matrix solution methods individually and then to combine the two methods into a hybrid scheme to achieve additional convergence rate benefits. One iterative matrix solver investigated is the sparse iterative method (SIM) which is a stationary, Jacobi-like solver but uses a sparse and not a banded matrix, with matrix elements corresponding to strong interactions, rather than position in the matrix. In this paper, the SIM is modified to include an adaptive relaxation scheme to improve its convergence speed and numerical stability. Another iterative scheme investigated is the nonstationary biconjugate gradient stabilized (BiCGSTAB) method. It is shown that the BiCGSTAB is considerably improved when the method is preconditioned by the sparse matrix used in the SIM method. Finally, a hybrid scheme is proposed which combines both SIM-AR and BiCGSTAB-precon and it is shown that the hybrid gives best results on the problems considered. Examples giving convergence time versus accuracy are presented for two problems: a wire-grid plate, and a wire-grid partial helicopter.