A fast adaptive algorithm for scattering from a two dimensional radially-symmetric potential

TL;DR

This paper introduces a fast, adaptive black box algorithm that efficiently solves 2D scattering problems with radially-symmetric potentials using FFTs and low-rank operator properties, applicable to both static and time-dependent cases.

Contribution

The paper presents a novel FFT-based method utilizing scattering matrices for efficient 2D scattering from radially-symmetric potentials, including extensions to time-dependent problems.

Findings

Efficient solution of scattering problems with singular and discontinuous potentials.

Demonstrated high accuracy and computational speed through numerical examples.

Extension of the method to time-dependent scattering problems.

Abstract

In the present paper we describe a simple black box algorithm for efficiently and accurately solving scattering problems related to the scattering of time-harmonic waves from radially-symmetric potentials in two dimensions. The method uses FFTs to convert the problem into a set of decoupled second-kind Fredholm integral equations for the Fourier coefficients of the scattered field. Each of these integral equations are solved using scattering matrices, which exploit certain low-rank properties of the integral operators associated with the integral equations. The performance of the algorithm is illustrated with several numerical examples including scattering from singular and discontinuous potentials. Finally, the above approach can be easily extended to time-dependent problems. After outlining the necessary modifications we show numerical experiments illustrating the performance of the algorithm in this setting.