A Robust Optimal Finite Difference Scheme for the Three-Dimensional Helmholtz Equation

Dongsheng Cheng,Baowen Chen,Xiangling Chen

doi:10.1155/2019/8532408

Abstract

We propose a robust optimal 27-point finite difference scheme for the Helmholtz equation in three-dimensional domain. In each direction, a special central difference scheme with 27 grid points is developed to approximate the second derivative operator. The 27 grid points are divided into four groups, and each group is involved in the difference scheme by the manner of weighted combination. As for the approximation of the zeroth-order term, we use the weighted average of all the 27 points, which are also divided into four groups. Finally, we obtain the optimal weights by minimizing the numerical dispersion with the least-square method. In comparison with the rotated difference scheme based on a staggered-grid method, the new scheme is simpler, more practical, and much more robust. It works efficiently even if the step sizes along different directions are not equal. However, rotated scheme fails in this situation. We also present the convergence analysis and dispersion analysis. Numerical examples demonstrate the effectiveness of the proposed scheme.

Highlights

Many of the physical problems are governed by the wave equation, which is well known as the Helmholtz equation in frequency domain
We point out that both the new scheme and rotated scheme are similar to the “dispersion relation preserving” (DRP) schemes that were originally introduced in [19] and continued with [20], since all of these methods are based on reducing the numerical dispersion error related to wave problems
We present the 27-point finite difference stencil with numbering in Figure 1, where the grid points are numbered as (m0, n0, l0) with m0, n0, l0 ∈ {−1, 0, 1}

Summary

Introduction

Many of the physical problems are governed by the wave equation, which is well known as the Helmholtz equation in frequency domain. To obtain a robust difference scheme for the 3D Helmholtz equation, in this paper, we propose a robust optimal 27-point difference scheme, which remains weighted and second order in accuracy but is rotation-free. We point out that both the new scheme and rotated scheme are similar to the “dispersion relation preserving” (DRP) schemes that were originally introduced in [19] and continued with [20], since all of these methods are based on reducing the numerical dispersion error related to wave problems.

A Robust Optimal 27-Point Finite Difference Scheme

Numerical Experiments

Conclusions