Multiple-scattering frequency-time hybrid solver for the wave equation in interior domains

TL;DR

This paper introduces a novel frequency-time hybrid solver for the wave equation in interior domains, combining multiple scattering, boundary integral equations, and Fourier transform techniques for accurate, efficient long-duration wave simulations.

Contribution

It presents a new hybrid approach that decomposes interior wave problems into scattering events and employs Fourier transforms for dispersionless, spectrally-accurate time evolution.

Findings

Demonstrates high accuracy in numerical examples

Achieves efficient long-time wave simulations

Regularizes the interior frequency domain problem

Abstract

This paper proposes a frequency-time hybrid solver for the time-dependent wave equation in two-dimensional interior spatial domains. The approach relies on four main elements, namely, 1) A multiple scattering strategy that decomposes a given interior time-domain problem into a sequence of limited-duration time-domain problems of scattering by overlapping open arcs, each one of which is reduced (by means of the Fourier transform) to a sequence of Helmholtz frequency-domain problems; 2) Boundary integral equations on overlapping boundary patches for the solution of the frequency-domain problems in point 1); 3) A smooth "Time-windowing and recentering" methodology that enables both treatment of incident signals of long duration and long time simulation; and, 4) A Fourier transform algorithm that delivers numerically dispersionless, spectrally-accurate time evolution for given incident fields. By recasting the interior time-domain problem in terms of a sequence of open-arc multiple scattering events, the proposed approach regularizes the full interior frequency domain problem-which, if obtained by either Fourier or Laplace transformation of the corresponding interior time-domain problem, must encapsulate infinitely many scattering events, giving rise to non-uniqueness and eigenfunctions in the Fourier case, and ill conditioning in the Laplace case. Numerical examples are included which demonstrate the accuracy and efficiency of the proposed methodology.