A dispersion minimizing scheme for the 3-D Helmholtz equation based on ray theory

TL;DR

This paper introduces a novel dispersion minimizing finite difference scheme for the 3-D Helmholtz equation, leveraging ray theory to achieve minimal phase errors and efficient large-scale wave simulations.

Contribution

A new compact finite difference scheme based on ray theory that significantly reduces dispersion errors in 3-D Helmholtz problems, enabling accurate solutions with fewer grid points per wavelength.

Findings

Phase errors are very small, comparable to advanced FEM methods.

Accurate solutions achieved with only 5-6 points per wavelength.

Effective as a coarse discretization in multigrid methods, reducing computational costs.

Abstract

We develop a new dispersion minimizing compact finite difference scheme for the Helmholtz equation in 2 and 3 dimensions. The scheme is based on a newly developed ray theory for difference equations. A discrete Helmholtz operator and a discrete operator to be applied to the source and the wavefields are constructed. Their coefficients are piecewise polynomial functions of $hk$, chosen such that phase and amplitude errors are minimal. The phase errors of the scheme are very small, approximately as small as those of the 2-D quasi-stabilized FEM method and substantially smaller than those of alternatives in 3-D, assuming the same number of gridpoints per wavelength is used. In numerical experiments, accurate solutions are obtained in constant and smoothly varying media using meshes with only five to six points per wavelength and wave propagation over hundreds of wavelengths. When used as a coarse level discretization in a multigrid method the scheme can even be used with downto three points per wavelength. Tests on 3-D examples with up to $10^8$ degrees of freedom show that with a recently developed hybrid solver, the use of coarser meshes can lead to corresponding savings in computation time, resulting in good simulation times compared to the literature.