Convolution quadrature methods for time-domain scattering from unbounded penetrable interfaces

TL;DR

This paper introduces a novel boundary integral equation approach combining Convolution Quadrature and Windowed Green Function methods for efficient and accurate simulation of time-domain scattering problems involving unbounded penetrable interfaces in two dimensions.

Contribution

It develops a new numerical framework that transforms unbounded interface scattering problems into manageable bounded problems using CQ and WGF, enabling high-order accurate solutions.

Findings

Super-algebraic error decay with increasing window size

Effective solution of complex frequency-domain transmission problems

Numerical demonstrations in waveguides and layered media

Abstract

This paper presents a class of boundary integral equation methods for the numerical solution of acoustic and electromagnetic time-domain scattering problems in the presence of unbounded penetrable interfaces in two-spatial dimensions. The proposed methodology relies on Convolution Quadrature (CQ) methods in conjunction with the recently introduced Windowed Green Function (WGF) method. As in standard time-domain scattering from bounded obstacles, a CQ method of the user's choice is utilized to transform the problem into a finite number of (complex) frequency-domain problems posed on the domains involving penetrable unbounded interfaces. Each one of the frequency-domain transmission problems is then formulated as a second-kind integral equation that is effectively reduced to a bounded interface by means of the WGF method---which introduces errors that decrease super-algebraically fast as the window size increases. The resulting windowed integral equations can then be solved by means of any (accelerated or unaccelerated) off-the-shelf Helmholtz boundary integral equation solver capable of handling complex wavenumbers with a large imaginary part. A high-order Nystr\"om method based on Alpert quadrature rules is utilized here. A variety of numerical examples including wave propagation in open waveguides as well as scattering from multiply layered media demonstrate the capabilities of the proposed approach.