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

• Published in: Journal of computational physics Vol. 228; no. 1; pp. 75 - 95
• Main Authors: Kurkcu, Harun, Reitich, Fernando
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier Inc 10.01.2009; Elsevier
• Subjects: Algorithms; Computational techniques; Exact sciences and technology; Free-space; Green's functions; Green’s function periodic domain; Helmholtz equation; High frequencies; High-frequency; Integrals; Mathematical analysis; Mathematical methods in physics; Physics; Quadratures; Representations; Rough-surface; Free-space; Helmholtz equation; Green’s function periodic domain; Rough-surface; High-frequency; Helmholtz equations; Periodic function; Numerical method; Green function; Calculation methods; Algorithms; Calculation; Integral representation; Green's function periodic domain
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2008.08.021

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” G qp. 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 G qp 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 G qp 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.