Higher-order accurate two-step finite difference schemes for the many-dimensional wave equation

The accurate simulation of wave propagation is a problem of longstanding interest. In this article, the focus is on higher-order accurate finite difference schemes for the wave equation in any number of spatial dimensions. In particular, two-step schemes (which operate over three time levels) are studied. A novel approach to the construction of schemes exhibiting both isotropy and accuracy is presented using modified equation techniques, and allowing for the specification of precise stencils of operation for the scheme, and thus direct control over stencil size and thus operation counts per time step. Both implicit and explicit schemes are presented, as well as parameterised families of such schemes under conditions specifying the order of isotropy and accuracy. Such conditions are framed in terms of a set of coupled constraints which are nonlinear in general, but linear for a fixed Courant number. Depending on the particular choice of stencils, it is often possible to develop schemes for which the traditional Courant-Friedrichs-Lewy condition is exceeded. A wide variety of families of such schemes is presented in one, two and three spatial dimensions, and accompanied by illustrations of numerical dispersion as well as convergence results confirming higher-order accuracy up to eighth order.