A dispersion minimizing finite difference scheme for the Helmholtz equation based on point-weighting

In this paper, we develop a new dispersion minimizing finite difference scheme for the Helmholtz equation with perfectly matched layer (PML) in two dimensional domain, which is a second order 9-point scheme. To discretize the second derivative operator, we employ a linear combination of a point and its neighboring grid points to replace each of the five points in the traditional central difference scheme. Based on minimizing the numerical dispersion, the combination weights are determined by minimizing the numerical dispersion with a flexible selection strategy. The new scheme is simple, rotation-free, and pointwise consistent with the equation, which is different from the classical rotated 9-point difference scheme obtained by combining the Cartesian coordinate system and the rotated system. Moreover, it is a robust scheme even if the step sizes of different directions are not equal. Convergence analysis and dispersion analysis are given. Several numerical examples are presented to illustrate the numerical convergence and effectiveness of the new scheme.