Finite difference schemes and digital waveguide networks for the wave equation: stability, passivity, and numerical dispersion

In this paper, some simple families of explicit two-step finite difference methods for solving the wave equation in two and three spatial dimensions are examined. These schemes depend on several free parameters, and can be associated with so-called interpolated digital waveguide meshes. Special attention is paid to the stability properties of these schemes (in particular the bounds on the space-step/time-step ratio) and their relationship with the passivity condition on the related digital waveguide networks. Boundary conditions are also discussed. An analysis of the directional numerical dispersion properties of these schemes is provided, and minimally directionally-dispersive interpolated digital waveguide meshes are constructed.