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.

翻译：由于其解具有高度振荡特性，亥姆霍兹方程在数值求解上极具挑战性。为获得合理数值解，通常需要采用远小于波数倒数的网格尺寸（即污染效应）。高阶格式因能更有效抑制污染效应而备受青睐。本文针对具有奇异源项的二维亥姆霍兹方程，提出一种可处理矩形域上任意边界条件组合（狄利克雷、诺伊曼及阻抗边界）的高阶紧致有限差分方法。所提方法对常波数情形可实现六阶精度相容性，对分段常波数情形可达五阶精度相容性。为降低污染效应，我们提出基于平面波平均截断误差的新型污染最小化策略。数值实验表明，本文提出的降污染有限差分格式在多个关键性能指标上优于现有先进有限差分方案，特别是在波数k与网格尺寸h满足kh接近1的关键预渐近区域中展现出显著优势。