A quasilinear complexity algorithm for the numerical simulation of scattering from a two-dimensional radially symmetric potential

TL;DR

This paper introduces a novel algorithm for simulating scattering from radially symmetric potentials in 2D, achieving quasilinear complexity in the wavenumber, significantly improving over traditional quadratic-time solvers.

Contribution

The paper presents a new efficient solver specifically for radially symmetric potentials, reducing computational complexity from quadratic to near-linear in the wavenumber.

Findings

Solver operates in O(k log k) time for typical cases

Numerical experiments confirm efficiency and accuracy

Code is publicly available for use and further testing

Abstract

Standard solvers for the variable coefficient Helmholtz equation in two spatial dimensions have running times which grow quadratically with the wavenumber $k$. Here, we describe a solver which applies only when the scattering potential is radially symmetric but whose running time is $\mathcal{O}\left(k \log(k) \right)$ in typical cases. We also present the results of numerical experiments demonstrating the properties of our solver, the code for which is publicly available.