Iterative schemes for high order compact discretizations to the exterior Helmholtz equation^∗

¹ Department of Earth and Ocean Sciences, The University of British Columbia, 2329 West Mall Vancouver, BC V6T 124, Canada
² Currently at Mathematics Division, College of Sciences, Alfaisal University, P.O. Box 50927, Riyadh, 11533 Kingdom of Saudi Arabia
³ Department of Mathematics, Tel Aviv University, Tel Aviv, Israel
eliturkel@gmail.com

Received: 26 August 2009
Revised: 3 January 2011

Abstract

We consider high order finite difference approximations to the Helmholtz equation in an exterior domain. We include a simplified absorbing boundary condition to approximate the Sommerfeld radiation condition. This yields a large, but sparse, complex system, which is not self-adjoint and not positive definite. We discretize the equation with a compact fourth or sixth order accurate scheme. We solve this large system of linear equations with a Krylov subspace iterative method. Since the method converges slowly, a preconditioner is introduced, which is a Helmholtz equation but with a modified complex wavenumber. This is discretized by a second or fourth order compact scheme. The system is solved by BICGSTAB with multigrid used for the preconditioner. We study, both by Fourier analysis and computations this preconditioned system especially for the effects of high order discretizations.