Sixth Order Compact Finite Difference Method for 2D Helmholtz Equations with Singular Sources and Reduced Pollution Effect

TL;DR

This paper introduces a sixth order compact finite difference method for 2D Helmholtz equations with singular sources, effectively reducing pollution effects and handling various boundary conditions, thus improving numerical accuracy for highly oscillatory solutions.

Contribution

The paper presents a novel sixth order compact finite difference scheme with a pollution minimization strategy for 2D Helmholtz equations, enhancing accuracy and efficiency over existing methods.

Findings

Achieves sixth order consistency for constant wavenumber.

Reduces pollution effect compared to state-of-the-art schemes.

Performs well in the pre-asymptotic region where kh is near 1.

Abstract

Due to its highly oscillating solution, the Helmholtz equation is numerically challenging to solve. To obtain a reasonable solution, a mesh size that is much smaller than the reciprocal of the wavenumber is typically required (known as the pollution effect). High order schemes are desirable, because they are better in mitigating the pollution effect. In this paper, we present a high order compact finite difference method for 2D Helmholtz equations with singular sources, which can also handle any possible combinations of boundary conditions (Dirichlet, Neumann, and impedance) on a rectangular domain. Our method achieves a sixth order consistency for a constant wavenumber, and a fifth order consistency for a piecewise constant wavenumber. To reduce the pollution effect, we propose a new pollution minimization strategy that is based on the average truncation error of plane waves. Our numerical experiments demonstrate the superiority of our proposed finite difference scheme with reduced pollution effect to several state-of-the-art finite difference schemes, particularly in the critical pre-asymptotic region where $\textsf{k} h$ is near $1$ with $\textsf{k}$ being the wavenumber and $h$ the mesh size.