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.

求解亥姆霍兹方程的有效而精确的数值方案对于各种与波传播相关的反问题（例如全波形反演问题）的成功至关重要。但是，众所周知，多维Helmholtz方程的数值求解非常困难，尤其是当包含完全匹配层（PML）边界条件时。本文提出了在二维域中具有PML的Helmholtz方程的最优13点有限差分格式。提供了精确波数数值近似的误差分析。在误差分析的基础上，提出了最优的13点有限差分方案，以使数值离散最小。提出了两种选择最优参数的实用策略。