A new parallel algorithm with high-order finite difference scheme for solving the Helmholtz equation in two and three dimensions

Abstract

This paper reports a class of new hybrid compact finite-difference schemes with high-order accuracy for solving the Helmholtz equation in two and three dimensions. The innovation of the scheme is to hybridize explicit and implicit compact schemes to deal with the solution and its first- and second-order derivatives, and to solve it by a step-by-step coupled iterative method. In response to the inefficiency of serial algorithms in solving the Helmholtz equation with large wavenumber and the issue of memory overflow on a single processor making computation infeasible because of excessively large computational scale, a parallel algorithm is proposed based on the Message Passing Interface (MPI) environment. Truncation error analysis demonstrates sixth-order accuracy for the proposed scheme, and numerical experiments confirm the theoretical sixth-order accuracy for problems with variable and large wavenumber. Also, the MPI-based parallel algorithm exhibits great parallel speedup and enables the tackling of large-scale problems that are beyond the reach of serial algorithms.