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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 954658, 14 pages
http://dx.doi.org/10.1155/2014/954658
Research Article

Multigrid Method for Solution of 3D Helmholtz Equation Based on HOC Schemes

1Department of Mathematics, Abdul Wali Khan University Mardan, Pakistan
2Department of Basic Sciences, UET, Peshawar, Pakistan

Received 22 March 2014; Revised 17 July 2014; Accepted 17 July 2014; Published 27 August 2014

Academic Editor: Sher Afzal Khan

Copyright © 2014 Fazal Ghaffar et al. 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

A higher order compact difference (HOC) scheme with uniform mesh sizes in different coordinate directions is employed to discretize a two- and three-dimensional Helmholtz equation. In case of two dimension, the stencil is of 9 points while in three-dimensional case, the scheme has 27 points and has fourth- to fifth-order accuracy. Multigrid method using Gauss-Seidel relaxation is designed to solve the resulting sparse linear systems. Numerical experiments were conducted to test the accuracy of the sixth-order compact difference scheme with Multigrid method and to compare it with the standard second-order finite-difference scheme and fourth-order compact difference scheme. Performance of the scheme is tested through numerical examples. Accuracy and efficiency of the new scheme are established by using the errors norms .