Sixth-Order Compact Finite Difference Method for 2D Helmholtz Equations with Singular Sources and Reduced Pollution Effect (Communications in Computational Physics, 2023)
Due to its highly oscillating solution, the Helmholtz equation is numericallychallenging to solve. To obtain a reasonable solution, a mesh size that is much smallerthan the reciprocal of the wavenumber is typically required (known as the pollutioneffect). 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 methodfor 2D Helmholtz equations with singular sources, which can also handle any possiblecombinations of boundary conditions (Dirichlet, Neumann, and impedance) on a rectangular domain. Our method is sixth-order consistent for a constant wavenumber, andfifth-order consistent for a piecewise constant wavenumber. To reduce the pollution effect, we propose a new pollution minimization strategy that is based on the averagetruncation error of plane waves. Our numerical experiments demonstrate the superiority of our proposed finite difference scheme with reduced pollution effect to severalstate-of-the-art finite difference schemes, particularly in the critical pre-asymptotic region where $kh$ is near 1 with $k$ being the wavenumber and $h$ the mesh size.
Keywords: Helmholtz equation; finite difference; pollution effect; interface; pollution minimization; mixed boundary conditions; corner treatment.