Finite-Difference Methods for the Wave Equation with Reduced Dispersion Errors

Abstract

A new methodology was proposed in Finkelstein and Kastner (2007,2008) to derive finite-difference (FD) schemes in the joint time-space domain to reduce dispersion error. The key idea is that the true dispersion relation is satisfied exactly at some specified wavenum- bers. Liu and Sen (2009) further developed their idea, going to 2D and 3D. In our work, we will prove that the system for coefficients of these new schemes is solvable for any normalized wavenumbers up to the Nyquist. We will also look at the system matrix and prove that we can get higher order approximation to the dispersion at arbitrary normalized wavenumbers up to the Nyquist.