Dispersion Free Finite Difference Scheme for Time Harmonic Wave Equation

Abstract

Dispersion Free Finite Difference Scheme for Time Harmonic Wave Equation In this work, we construct novel central finite difference schemes for solving the Helmholtz equation by following the idea of Nehrbass and Yau Shu in one and two dimensions. We present an alternative construction by which not only central f inite difference schemes but all types of schemes (forward, backward) can be made exact for Helmholtz equation. We give explicit expressions for the coefficient of the central node in the standard central finite difference scheme of any order which makes the novel scheme dispersion free in one dimension and optimal in two dimensions. Such schemes add no cost for the implementation and one needs not to write brand new code instead just need to change the central node coefficient with the new one presented in this work to have highly accurate results. Comparison of numerical solutions obtained using novel schemes with standard schemes for Helmholtz equation in single and higher dimensions are given.