Solving exterior problems of wave propagation based on an iterative variation of local DtN operators

Abstract

This paper discusses the problem of wave reflection from the fictitious boundary, with particular regard to the higher harmonic modes. This problem occurs when solving the wave equation in exterior domains using an asymptotic local low-order Dirichlet-to-Neumann (DtN) map for computational procedures applied to a finite domain. We demonstrate that the amplitudes of the reflected fictitious harmonics depend on the wave number, the location of the fictitious boundary, as well as on the local DtN operator used in the computations. Moreover, we show that a constant value of the asymptotic local low-order operator cannot sufficiently eliminate the amplitudes of all reflected waves, and that the results are poor especially for higher harmonics. We propose therefore an iterative method, which varies the tangential dependence of the local operator in each computational step. We only discuss some logical and interesting choices for the operators although this method permits several possibilities on how to vary the operator. The method is simple to apply and the presented examples demonstrate that the accuracy is considerably improved by iterations.