Efficient time-domain scattering synthesis via frequency-domain singularity subtraction

TL;DR

This paper introduces a frequency-domain singularity subtraction method to improve the efficiency of time-domain scattering simulations involving trapping obstacles by accurately handling complex resonances.

Contribution

It presents a novel regularization technique that isolates and computes resonance contributions efficiently using rational approximation and adaptive algorithms.

Findings

Significantly accelerates inverse Fourier transform convergence in trapping obstacle scattering.

Accurately identifies complex resonances and their residues.

Demonstrates improved computational efficiency in numerical experiments.

Abstract

Fourier transform-based methods enable accurate, dispersion-free simulations of time-domain scattering problems by evaluating solutions to the Helmholtz equation at a discrete set of frequencies sufficient to approximate the inverse Fourier transform. However, in the case of scattering by trapping obstacles, the Helmholtz solution exhibits nearly-real complex resonances -- which significantly slows the convergence of numerical inverse transform. To address this difficulty this paper introduces a frequency-domain singularity subtraction technique that regularizes the integrand of the inverse transform and efficiently computes the singularity contribution via a combination of a straightforward and inexpensive numerical technique together with a large-time asymptotic expansion. Crucially, all relevant complex resonances and their residues are determined via rational approximation of integral equation solutions at real frequencies. An adaptive algorithm is employed to ensure that all relevant complex resonances are properly identified.