Interpolating Wavelets and Adaptive Finite Difference Schemes for Solving Maxwell's Equations: The Effects of Gridding

In this paper, we discuss the use of the sparse point representation (SPR) methodology for adaptive finite-difference simulations in computational eletromagnetics. The principle of the SPR method is to represent the solution only through those point values indicated by the significant wavelet coefficients, which are used as local regularity indicators. Recently, two kinds of SPR schemes have been considered for solving Maxwell's equations: 1) staggered grids in the time-space domain are used for the discretization of the magnetic and electrical fields, as in the finite-differences time-domain (FDTD) scheme and 2) nonstaggered grids are used in combination with Runge-Kutta ODE solvers. In both cases, 1-D simulations of the SPR method leads to sparse grids that adapt in space to the local smoothness of the fields, and, at the same time, track the evolution of the fields over time with substantial gain in memory and computational speed. However, in the latter case, we found spurious oscillations in the simulations. Therefore, before extending the implementation of the SPR method to higher dimensions, we wanted to evaluate which of these two SPR strategies is more convenient. After a careful theoretical analysis of stability and numerical dispersion comparing the schemes in staggered and nonstaggered grids, we conclude that schemes for staggered grids seem to be preferable from the dispersion viewpoint, especially for low-order schemes and coarse grids. However, by adapting the grid density and increasing the order, SPR schemes for nonstaggered grids also show good performance. In our experiments, no spurious oscillations were detected. We observed that, for a given accuracy, the adaptive scheme on a nonstaggered grid requires less computational effort. Since the use of nonstaggered grids increases the stability range and facilitates the implementation of adaptive strategies, we believe that the SPR method in nonstaggered grids has a very good potential for computational electromagnetics.