Sixth-order finite difference scheme for the Helmholtz equation with inhomogeneous Robin boundary condition

Yang Zhang,Rui Guo,Kun Wang

doi:10.1186/s13662-019-2304-0

Yang Zhang, Rui Guo + Show 1 more

Open Access

https://doi.org/10.1186/s13662-019-2304-0

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Abstract

In this paper, a class of sixth-order finite difference schemes for the Helmholtz equation with inhomogeneous Robin boundary condition is derived. This scheme is based on the sixth-order approximation for the Robin boundary condition by using the Helmholtz equation and the Taylor expansion, by which the ghost points in the scheme on the domain can be eliminated successfully. Some numerical examples are shown to verify its correctness and robustness with respect to the wave number.

Highlights

1 Introduction In this paper, we focus on the Helmholtz equation with inhomogeneous Robin boundary condition on four boundaries
Various numerical methods were developed in the past decades, such as the finite difference method, the finite element method, the boundary element method, and other techniques
By applying the one-sided approximation of the derivative and Taylor expansion carefully, we derive the sixth-order scheme for the inhomogeneous Robin boundary condition (5), by which the ghost points in the scheme on the domain can be eliminated successfully

Summary

Introduction

We focus on the Helmholtz equation with inhomogeneous Robin boundary condition on four boundaries. When the wave number k is large, the solution of the problem becomes highly oscillating and efficient numerical methods are required in order to get high performance simulation results In this topic, various numerical methods were developed in the past decades, such as the finite difference method (see, e.g., [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]), the finite element method (see, e.g., [17,18,19,20,21,22,23,24,25]), the boundary element method (see, e.g., [26,27,28,29,30]), and other techniques (see, e.g., [17, 31,32,33]). The body force f is equal to zero, we simplify the Helmholtz equation with an inhomogeneous Robin boundary condition as follows:. Some numerical experiments are shown to verify the correctness and the robustness of the scheme with respect to the wave number, too

Numerical scheme

Conclusions