Fourier beyond dispersion: Wavenumber explicit and precise accuracy of FDMs for the Helmholtz equation

Hui Zhang

doi:10.1016/j.aml.2025.109576

Fourier beyond dispersion: Wavenumber explicit and precise accuracy of FDMs for the Helmholtz equation

Hui Zhang

Xi'an Jiaotong-Liverpool University

Research output: Contribution to journal › Article › peer-review

Abstract

We propose a practical tool for evaluating and comparing the accuracy of FDMs for the Helmholtz equation. The tool based on Fourier analysis makes it easy to find wavenumber explicit order of convergence, and can be used for rigorous proof. It fills in the gap between the dispersion analysis and the actual error with source term. We illustrate it for classical and some dispersion free schemes in 1D, with conclusions verified by numerical experiments.

Original language	English
Article number	109576
Journal	Applied Mathematics Letters
Volume	168
DOIs	https://doi.org/10.1016/j.aml.2025.109576
Publication status	E-pub ahead of print - Apr 2025

Keywords

Dispersion analysis
Finite difference
Fourier analysis
Helmholtz equation
Pollution effect

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10.1016/j.aml.2025.109576

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abstract = "We propose a practical tool for evaluating and comparing the accuracy of FDMs for the Helmholtz equation. The tool based on Fourier analysis makes it easy to find wavenumber explicit order of convergence, and can be used for rigorous proof. It fills in the gap between the dispersion analysis and the actual error with source term. We illustrate it for classical and some dispersion free schemes in 1D, with conclusions verified by numerical experiments.",

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AB - We propose a practical tool for evaluating and comparing the accuracy of FDMs for the Helmholtz equation. The tool based on Fourier analysis makes it easy to find wavenumber explicit order of convergence, and can be used for rigorous proof. It fills in the gap between the dispersion analysis and the actual error with source term. We illustrate it for classical and some dispersion free schemes in 1D, with conclusions verified by numerical experiments.

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KW - Finite difference

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KW - Pollution effect

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Fourier beyond dispersion: Wavenumber explicit and precise accuracy of FDMs for the Helmholtz equation

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Keywords

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