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Is Pollution Effect of Finite Difference Schemes Avoidable for Multi-Dimensional Helmholtz Equations with High Wave Numbers?

Part of: Partial differential equations, boundary value problems

Published online by Cambridge University Press:  07 February 2017

Kun Wang  and
Yau Shu Wong
Kun Wang*
Affiliation:
College of Mathematics and Statistics, Chongqing University, Chongqing 401331, P.R. China Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton T6G 2G1, Canada Institute of Computing and Data Sciences, Chongqing University, Chongqing 400044, P.R. China
Yau Shu Wong*
Affiliation:
Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton T6G 2G1, Canada
*
*Corresponding author.Email addresses:kunwang@cqu.edu.cn, wk580@163.com (K.Wang), yaushu.wong@ualberta.ca (Y. S. Wong)
*Corresponding author.Email addresses:kunwang@cqu.edu.cn, wk580@163.com (K.Wang), yaushu.wong@ualberta.ca (Y. S. Wong)
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Abstract

This paper presents an approach using the method of separation of variables applied to 2D Helmholtz equations in the Cartesian coordinate. The solution is then computed by a series solutions resulted from solving a sequence of 1D problems, in which the 1D solutions are computed using pollution free difference schemes. Moreover, non-polluted numerical integration formulae are constructed to handle the integration due to the forcing term in the inhomogeneous 1D problems. Consequently, the computed solution does not suffer the pollution effect. Another attractive feature of this approach is that a direct method can be effectively applied to solve the tridiagonal matrix resulted from numerical discretization of the 1D Helmholtz equation. The method has been tested to compute 2D Helmholtz solutions simulating electromagnetic scattering from an open large cavity and rectangular waveguide.

Keywords

Helmholtz equationseparation of variableshigh wave numberpollution free finite ifference schemelarge cavity problemrectangular waveguide

MSC classification

Secondary: 65N06: Finite difference methods 65N15: Error bounds 65N22: Solution of discretized equations
Type
Research Article
Information
Communications in Computational Physics , Volume 21 , Issue 2 , February 2017 , pp. 490 - 514
Copyright
Copyright © Global-Science Press 2017 

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