A high-order compact scheme for the one-dimensional Helmholtz equation with a discontinuous coefficient

Abstract

In this paper, based on the idea of the immersed interface method, a fourth-order compact finite difference scheme is proposed for solving one-dimensional Helmholtz equation with discontinuous coefficient, jump conditions are given at the interface. The Dirichlet boundary condition and the Neumann boundary condition are considered. The Neumann boundary condition is treated with a fourth-order scheme. Numerical experiments are included to confirm the accuracy and efficiency of the proposed method.

Keywords:

• the immersed interface method
• Helmholtz equation
• discontinuous coefficient
• compact finite difference scheme

2010 AMS Subject Classifications :

• 65N06
• 65N50

Acknowledgements

The author thanks Prof. Zhilin Li for many helpful discussions. This research is supported by the National Natural Science Foundation of China Grant (Nos. 10972058 and 11161036), the Key Project of the Chinese Ministry of Education Grant (No. 209134), and the National Natural Science Foundation of Ningxia Grant (No. NZ1051).