Application of the Difference Gaussian Rules to Solution of Hyperbolic Problems

Two of the authors earlier suggested a method of calculating special grid steps for three point finite-difference schemes which yielded exponential superconvergence of the Neumann-to-Dirichlet map. We apply this approach to solve the two-dimensional time-domain wave problem and the 2.5-D elasticity system in cylindrical coordinates. Our numerical experiments exhibit exponential convergence at prescribed points, with the cost per grid node close to that of the standard second order finite-difference scheme. The scheme demonstrates high accuracy with slightly more than two grid points per wavelength. The reduction of the grid size by one order compared to the standard scheme with the equidistant grids is observed.