A crucial part of successful wave propagation related inverse problems is an efficient and accurate numerical scheme for solving the seismic wave equations. In particular, the numerical solution to a multi-dimensional Helmholtz equation can be troublesome when the perfectly matched layer (PML) boundary condition is implemented. In this paper, we present a general approach for constructing fourth-order finite difference schemes for the Helmholtz equation with PML in the two-dimensional domain based on point-weighting strategy. Particularly, we develop two optimal fourth-order finite difference schemes, optimal point-weighting 25p and optimal point-weighting 17p. It is shown that the two schemes are consistent with the Helmholtz equation with PML. Moreover, an error analysis for the numerical approximation of the exact wavenumber is provided. Based on minimizing the numerical dispersion, we implement the refined choice strategy for selecting optimal parameters and present refined point-weighting 25p and refined point-weighting 17p finite difference schemes. Furthermore, three numerical examples are provided to illustrate the accuracy and effectiveness of the new methods in reducing numerical dispersion.

成功的与波传播有关的反问题的关键部分是求解地震波方程的有效且精确的数值方案。尤其是，当实现完全匹配层（PML）边界条件时，多维Helmholtz方程的数值解会很麻烦。在本文中，我们提出了一种基于点加权策略在二维域中用PML构造Helmholtz方程的四阶有限差分格式的通用方法。特别是，我们开发了两个最优的四阶有限差分方案，最优点加权25p和最优点加权17p。结果表明，两种方案与带有PML的Helmholtz方程是一致的。此外，提供了对精确波数的数值近似的误差分析。在最小化数值离散的基础上，我们实施了用于选择最佳参数的改进选择策略，并提出了改进的点加权25p和改进的点加权17p有限差分方案。此外，提供了三个数值示例来说明新方法在减少数值离散方面的准确性和有效性。