Year: 2014
Author: K. Wang, Y. S. Wong
International Journal of Numerical Analysis and Modeling, Vol. 11 (2014), Iss. 4 : pp. 787–815
Abstract
In this paper, we develop pollution-free finite difference schemes for solving the non-homogeneous Helmholtz equation in one dimension. A family of high-order algorithms is derived by applying the Taylor expansion and imposing the conditions that the resulting finite difference schemes satisfied the original equation and the boundary conditions to certain degrees. The most attractive features of the proposed schemes are: first, the new difference schemes have a $2n$-order of rate of convergence and are pollution-free. Hence, the error is bounded even for the equation at high wave numbers. Secondly, the resulting difference scheme is simple, namely it has the same structure as the standard three-point central differencing regardless of the order of accuracy. Convergence analysis is presented, and numerical simulations are reported for the non-homogeneous Helmholtz equation with both constant and varying wave numbers. The computational results clearly confirm the superior performance of the proposed schemes.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/2014-IJNAM-552
International Journal of Numerical Analysis and Modeling, Vol. 11 (2014), Iss. 4 : pp. 787–815
Published online: 2014-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 29
Keywords: Helmholtz equation Finite difference method Convergence analysis High wave number Pollution-free High-order schemes.