Application of the Difference Gaussian Rules to Solution of Hyperbolic Problems: II. Global Expansion

This work is the sequel to S. Asvadurov et al. (2000, J. Comput. Phys.158, 116), where we considered a grid refinement approach for second-order finite-difference time domain schemes. This approach permits one to compute solutions of certain wave equations with exponential superconvergence. An algorithm was presented that generates a special sequence of grid steps, called “optimal”, such that a standard finite-difference discretization that uses this grid produces an accurate approximation to the Neumann-to-Dirichlet map. It was demonstrated that the application of this approach to some problems in, e.g., elastodynamics results in a computational cost that is an order of magnitude lower than that of the standard scheme with equally spaced gridnodes, which produces the same accuracy. The main drawback of the presented approach was that the accurate solution could be obtained only at some a priori selected points (receivers). Here we present an algorithm that, given a solution on the coarse “optimal” grid, accurately reconstructs the solution of the corresponding fine equidistant grid with steps that are approximately equal to the minimal step of the optimal (strongly nonuniform) grid. This “expansion” algorithm is based on postprocessing of the approximate solution, is local in time (but not in space), and has a cost comparable to that of the discrete Fourier transform. An approximate inverse to the “expansion” procedure—the “reduction” algorithm—is also presented. We show different applications of the developed procedures, including refinement of a nonmatching grid. Numerical examples for scalar wave propagation and 2.5D cylindrical elasticity are presented.