An efficient high-order Nystr\"om scheme for acoustic scattering by inhomogeneous penetrable media with discontinuous material interface

TL;DR

This paper introduces a fast, high-order Nyström method for solving acoustic scattering problems involving inhomogeneous media with discontinuous interfaces, combining analytical singularity resolution and FFT acceleration for efficiency.

Contribution

It presents a novel high-order Nyström scheme that efficiently handles discontinuous material interfaces in acoustic scattering using specialized quadratures and FFT-based acceleration.

Findings

Achieves $O(N \,\log N)$ computational complexity.

Demonstrates high accuracy in numerical experiments.

Provides a robust method for inhomogeneous media with discontinuities.

Abstract

This text proposes a fast, rapidly convergent Nystr\"{o}m method for the solution of the Lippmann-Schwinger integral equation that mathematically models the scattering of time-harmonic acoustic waves by inhomogeneous obstacles, while allowing the material properties to jump across the interface. The method works with overlapping coordinate charts as a description of the given scatterer. In particular, it employs "partitions of unity" to simplify the implementation of high-order quadratures along with suitable changes of parametric variables to analytically resolve the singularities present in the integral operator to achieve desired accuracies in approximations. To deal with the discontinuous material interface in a high-order manner, a specialized quadrature is used in the boundary region. The approach further utilizes an FFT based strategy that uses equivalent source approximations to accelerate the evaluation of large number of interactions that arise in the approximation of the volumetric integral operator and thus achieves a reduced computational complexity of $O(N \log N)$ for an $N$-point discretization. A detailed discussion on the solution methodology along with a variety of numerical experiments to exemplify its performance in terms of both speed and accuracy are presented in this paper.