A parallel high-order accuracy algorithm for the Helmholtz equations

Abstract

The numerical solution of the Helmholtz equations is challenging to compute when the wave numbers contained in the governing equation are large. In this paper, we present a parallel algorithm for this problem. A class of sixth-order hybrid compact finite-difference schemes for the Helmholtz equations is presented based on the Taylor expansion. To improve the efficiency of solving the large-wave-number problem, we implemented a parallel algorithm based on the Message Passing Interface environment to solve the discrete system. The validity and accuracy of the proposed method are verified by numerical examples. The method is also applicable to solving problems with oscillatory solutions, which are characterized by numerical instability as the wave number increases.

Keywords:

• Helmholtz equation
• sixth-order hybrid compact finite-difference scheme
• Parallel algorithm
• MPI

AMS Subject Classifications:

• 35J05
• 65Y05

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data Availability Statements

The datasets generated during and/or analyzed during the current study are available from the corresponding authors on reasonable request.

Additional information

Funding

This work was supported by the National Natural Science Foundation of China (Grant No. 11961054).