A generalized optimal fourth-order finite difference scheme for a 2D Helmholtz equation with the perfectly matched layer boundary condition

TL;DR

This paper introduces two new fourth-order finite difference schemes for solving the 2D Helmholtz equation with PML boundary conditions, improving accuracy and reducing numerical dispersion in wave propagation simulations.

Contribution

The paper develops two optimal point-weighting fourth-order finite difference schemes for the Helmholtz equation with PML, including error analysis and parameter refinement strategies.

Findings

The schemes achieve high accuracy in numerical experiments.

Refined parameter strategies further reduce numerical dispersion.

Numerical examples demonstrate improved solution quality.

Abstract

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.