Stable and efficient evaluation of periodized Green’s functions for the Helmholtz equation at high frequencies

A difficulty that arises in the context of infinite d-periodic rough-surface scattering relates to the effective numerical evaluation of the corresponding “quasi-periodic Green function” Gqp. Due to its relevance in a variety of applications, this problem has generated significant interest over the last 40 years, and a variety of numerical methods have been devised for this purpose. None of these methods to evaluate Gqp however, were designed for high-frequency calculations. As a result, in this regime, these methods become prohibitively expensive and/or unstable. Here we present a novel scheme that can be shown to outperform every alternative numerical evaluation procedure and is especially effective for high-frequency calculations. Our new algorithm is based on the use of some exact integrals that arise on judicious manipulation of the integral representation of Gqp and which reduce the overall problem to that of evaluation of a sequence of simpler integrals that can be effectively handled by standard quadrature formulas. We include a variety of numerical results that confirm that, indeed, our algorithm compares favorably with alternative methods.