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

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.