Two polynomial methods of calculating functions of symmetric matrices
USSR Computational Mathematics and Mathematical Physics
A perfectly matched layer for the absorption of electromagnetic waves
Journal of Computational Physics
Computational Mathematics and Mathematical Physics
The symmetric eigenvalue problem
The symmetric eigenvalue problem
Application of the difference Gaussian rules to solution of hyperbolic problems
Journal of Computational Physics
SIAM Journal on Numerical Analysis
Compensated optimal grids for elliptic boundary-value problems
Journal of Computational Physics
Reflectionless truncation of target area for axially symmetric anisotropic elasticity
Journal of Computational and Applied Mathematics
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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.