Hexagonal vs. Rectilinear Grids for Explicit Finite Difference Schemes for the Two-dimensional Wave Equation

Abstract

Finite difference schemes for the 2-D wave equation operating on hexagonal grids and the accompanyingnumerical dispersion properties have received little attention in comparison to schemes operating on rectilinear grids. This paper considers the hexagonal tiling of the wavenumber plane in order to show that thehexagonal grid is a more natural choice to emulate the isotropy of the Laplacian operator and the wave equation. Performance of the 7-point scheme on a hexagonal grid is better than previously reported as long as thecorrect stability limit and tiling of the wavenumber plane are taken into account. Numerical dispersion isanalysed as a function of temporal frequency to demonstrate directional cutoff frequencies. A comparison to9-point compact explicit schemes on rectilinear grids is presented using metrics relevant to acoustical simulation. It is shown that the 7-point hexagonal scheme has better computational efficiency than parameterised9-point compact explicit rectilinear schemes. A novel multiply-free 7-point hexagonal scheme is introducedand the 4-point scheme on a honeycomb grid is discussed