Numerical simulation of wave propagation in inhomogeneous media using Generalized Plane Waves

¹ Courant Institute of Mathematical Sciences, New York University, 251 Mercer street, New York, NY 10012, USA.
imbertgerard@cims.nyu.edu
² Department of Mathematical Sciences, University of Delaware, Newark DE 19716, USA.
monk@udel.edu

Received: 25 November 2015
Revised: 3 June 2016
Accepted: 12 October 2016

Abstract

The Trefftz Discontinuous Galerkin (TDG) method is a technique for approximating the Helmholtz equation (or other linear wave equations) using piecewise defined local solutions of the equation to approximate the global solution. When coefficients in the equation (for example, the refractive index) are piecewise constant it is common to use plane waves on each element. However when the coefficients are smooth functions of position, plane waves are no longer directly applicable. In this paper we show how Generalized Plane Waves (GPWs) can be used in a modified TDG scheme to approximate the solution for piecewise smooth coefficients in two dimensions. GPWs are approximate solutions to the equation that reduce to plane waves when the medium through which the wave propagates is constant. We shall show how to modify the TDG sesquilinear form to allow us to prove convergence of the GPW based version. The new scheme retains the high order convergence of the original TDG scheme (when the solution is smooth) and also retains the same number of degrees of freedom per element (corresponding to the directions of the GPWs). Unfortunately it looses the advantage that only skeleton integrals need to be performed. Besides proving convergence, we provide numerical examples to test our theory.