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An optimal 13-point finite difference scheme for a 2D Helmholtz equation with a perfectly matched layer boundary condition

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Abstract

Efficient and accurate numerical schemes for solving the Helmholtz equation are critical to the success of various wave propagation–related inverse problems, for instance, the full-waveform inversion problem. However, the numerical solution to a multi-dimensional Helmholtz equation is notoriously difficult, especially when a perfectly matched layer (PML) boundary condition is incorporated. In this paper, an optimal 13-point finite difference scheme for the Helmholtz equation with a PML in the two-dimensional domain is presented. An error analysis for the numerical approximation of the exact wavenumber is provided. Based on error analysis, the optimal 13-point finite difference scheme is developed so that the numerical dispersion is minimized. Two practical strategies for selecting optimal parameters are presented. Several numerical examples are solved by the new method to illustrate its accuracy and effectiveness in reducing numerical dispersion.

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Funding

The work is supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) through the individual Discovery Grant (RGPIN-2019-04830). The first author is also supported by the Alberta Innovates Graduate Student Scholarship that he received during his PhD study.

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  1. Department of Mathematics & Statistics, University of Calgary, AB, T2N 1N4, Calgary, Canada

    Hatef Dastour & Wenyuan Liao

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  1. Hatef Dastour
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  2. Wenyuan Liao
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Correspondence to Hatef Dastour.

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Dastour, H., Liao, W. An optimal 13-point finite difference scheme for a 2D Helmholtz equation with a perfectly matched layer boundary condition. Numer Algor 86, 1109–1141 (2021). https://doi.org/10.1007/s11075-020-00926-5

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  • DOI: https://doi.org/10.1007/s11075-020-00926-5

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