A new eighth-order implicit finite difference method to solve the three-dimensional Helmholtz equation

Abstract

This paper introduces a novel method for deducing high-order finite difference schemes for the three-dimensional Helmholtz equation, Δp±λ2p=f. The new method can reach an eighth-order of accuracy. It is based on an implicit formulation by computing simultaneously the unknown variable and its corresponding derivatives. The scheme is obtained from Taylor series expansion and plane wave theory analysis, and it is constructed from a few modifications to the standard centered finite differences. A classical dispersion analysis and the error between the numerical wavenumber and the exact wavenumber is given. Numerical experiments are presented to verify the feasibility and accuracy of the proposed methods for a broad range of λ-values. Moreover, the numerical method is capable to solve with great precision the monotone and oscillatory Helmholtz problem with variable λ-values. Thus, the proposed formulation results in an attractive method, easy to implement, with high order of accuracy but nearly the same computational cost as those of an explicit formulation.