Journal
APPLIED MATHEMATICS AND COMPUTATION
Volume 298, Issue -, Pages 221-235Publisher
ELSEVIER SCIENCE INC
DOI: 10.1016/j.amc.2016.11.015
Keywords
Helmholtz equation; High wave number; Local refinement; Polar and spherical coordinates; Pollution-free scheme
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Funding
- Natural Sciences and Engineering Research Council of Canada
- National Natural Science Foundation of China [11201506, 11501554, 11471312]
- Fundamental Research Funds for the Central Universities [106112015CDJXY100007]
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This paper presents effective difference schemes for solving the Helmholtz equation at high wave numbers. Considering the problems in the polar and spherical coordinates, we show that pollution-free difference schemes can be constructed for annulus and hollow sphere domains, but the pollution effect cannot be avoided in the neighborhood of the origin of the coordinate. Using the fact that the pollution effect appears in a small neighborhood of the origin of the coordinate, we propose a local mesh refinement approach so that a fine grid is applied only for a small region near the origin of the coordinate and a much larger grid size is employed in the remaining domain. The most attractive feature of this approach is that the computational storage can be significantly reduced while keeping almost the same numerical accuracy. Moreover, by applying a two-level local refinement technique, we can further enhance the performance of the proposed scheme. Numerical simulations are reported to verify the effectiveness of the proposed schemes for the Helmholtz problem in exterior domains. (C) 2016 Elsevier Inc. All rights reserved.
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