Compact High Order Accurate Schemes for the Three Dimensional Wave Equation

We construct a family of compact fourth order accurate finite difference schemes for the three dimensional scalar wave (d’Alembert) equation with constant or variable propagation speed. High order accuracy is of key importance for the numerical simulation of waves as it reduces the dispersion error (i.e., the pollution effect). The schemes that we propose are built on a stencil that has only three nodes in any coordinate direction or in time, which eliminates the need for auxiliary initial or boundary conditions. These schemes are implicit in time and conditionally stable. A particular scheme with the maximum Courant number can be chosen within the proposed class. The inversion at the upper time level is done by FFT for constant coefficients and multigrid for variable coefficients, which keeps the overall complexity of time marching comparable to that of a typical explicit scheme.