Table of Contents
ISRN Computational Mathematics
Volume 2014 (2014), Article ID 745849, 15 pages
http://dx.doi.org/10.1155/2014/745849
Research Article

Preconditioned Krylov Subspace Methods for Sixth Order Compact Approximations of the Helmholtz Equation

Mathematics Department, Idaho State University, 921 S 8th Avenue, Stop 8085, Pocatello, ID 83209-8085, USA

Received 5 August 2013; Accepted 10 November 2013; Published 21 January 2014

Academic Editors: L. Hajdu, S. Manservisi, Y. Peng, and R. Tuzun

Copyright © 2014 Yury Gryazin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider an efficient iterative approach to the solution of the discrete Helmholtz equation with Dirichlet, Neumann, and Sommerfeld-like boundary conditions based on a compact sixth order approximation scheme and lower order preconditioned Krylov subspace methodology. The resulting systems of finite-difference equations are solved by different preconditioned Krylov subspace-based methods. In the analysis of the lower order preconditioning developed here, we introduce the term “kth order preconditioned matrix” in addition to the commonly used “an optimal preconditioner.” The necessity of the new criterion is justified by the fact that the condition number of the preconditioned matrix in some of our test problems improves with the decrease of the grid step size. In a simple 1D case, we are able to prove this analytically. This new parameter could serve as a guide in the construction of new preconditioners. The lower order direct preconditioner used in our algorithms is based on a combination of the separation of variables technique and fast Fourier transform (FFT) type methods. The resulting numerical methods allow efficient implementation on parallel computers. Numerical results confirm the high efficiency of the proposed iterative approach.