ASYMPTOTIC DISPERSION CORRECTION IN GENERAL FINITE DIFFERENCE SCHEMES FOR HELMHOLTZ PROBLEMS.

Most numerical approximations of frequency-domain wave propagation problems suffer from the so-called dispersion error, which is the fact that plane waves at the discrete level oscillate at a frequency different from the continuous one. In this paper, we introduce a new technique to reduce the dispersion error in general finite difference (FD) schemes for frequency-domain wave propagation using the Helmholtz equation as a guiding example. Our method is based on the introduction of a shifted wavenumber in the FD stencil which we use to reduce the numerical dispersion for large enough numbers of grid points per wavelength (or for small enough meshsize), and thus we call the method asymptotic dispersion correction. The advantage of this technique is that the asymptotically optimal shift can be determined in closed form by computing the extrema of a function over a compact set. For one-dimensional Helmholtz equations, we prove that the standard 3-point stencil with shifted wavenumber does not have any dispersion error, and that the so-called pollution effect is completely suppressed. For higher dimensional Helmholtz problems, we give easy-to-use closed form formulas for the asymptotically optimal shift associated to the second-order 5-point scheme and a sixth-order 9-point scheme in two dimensions, and the 7-point scheme in three dimensions that yield substantially less dispersion error than their standard (unshifted) version. We illustrate this also with numerical experiments. [ABSTRACT FROM AUTHOR]