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Journal of Applied Mathematics
Volume 2014, Article ID 673106, 9 pages
http://dx.doi.org/10.1155/2014/673106
Research Article

Modified Finite Difference Schemes on Uniform Grids for Simulations of the Helmholtz Equation at Any Wave Number

1Department of Mathematics, COMSATS Institute of Information Technology, Lahore 54000, Pakistan
2Department of Mechanical Engineering, Faculty of Engineering, CECOS University of IT and Emerging Sciences, Peshawar 25000, Pakistan
3Department of Mathematics and Statistics, The University of Lahore, Lahore 54000, Pakistan
4Department of Mathematics, University College of Engineering, Sciences and Technology, Lahore Leads University, Lahore 54000, Pakistan

Received 11 April 2014; Accepted 20 June 2014; Published 12 August 2014

Academic Editor: Song Cen

Copyright © 2014 Hafiz Abdul Wajid 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

We construct modified forward, backward, and central finite difference schemes, specifically for the Helmholtz equation, by using the Bloch wave property. All of these modified finite difference approximations provide exact solutions at the nodes of the uniform grid for the second derivative present in the Helmholtz equation and the first derivative in the radiation boundary conditions for wave propagation. The most important feature of the modified schemes is that they work for large as well as low wave numbers, without the common requirement of a very fine mesh size. The superiority of the modified finite difference schemes is illustrated with the help of numerical examples by making a comparison with standard finite difference schemes.