An optimal 13-point finite difference scheme for a 2D Helmholtz equation with a perfectly matched layer boundary condition

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.