Fourier smoothed pre-corrected trapezoidal rule for solution of Lippmann-Schwinger integral equation

TL;DR

This paper introduces a Fourier smoothed pre-corrected trapezoidal rule for solving the Lippmann-Schwinger integral equation, achieving second-order convergence for problems with discontinuous material properties while maintaining computational efficiency.

Contribution

The authors propose a novel Nyström solver using Fourier smoothing that improves convergence for discontinuous densities and is simple to implement for complex geometries.

Findings

Achieves second-order convergence for discontinuous problems.

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

Demonstrates superior speed and accuracy in numerical experiments.

Abstract

For the numerical solution of the Lippmann-Schwinger equation, while the pre-corrected trapezoidal rule converges with high-order for smooth compactly supported densities, it exhibits only the linear convergence in the case of discontinuity in material properties across the interface. In this short article, we propose a Nystr\"{o}m solver based on "Fourier smoothed pre-corrected trapezoidal rule" that converges with second order for such scattering problems while maintaining the computational complexity of $O(N \log N)$. Moreover, the method is not only very simple to implement, it is also applicable to problems with geometrically complex inhomogeneities including those with corners and cusps. We present a variety of numerical experiments including comparative studies with competing approaches reported in [J. Comput. Phys., 200(2) (2004), 670--694] by Bruno and Hyde, and in [J. Fourier Anal. Appl., 11(4) (2005), 471-487 ] by Andersson and Holst to exemplify its performance in terms of speed and accuracy. This Fourier smoothed numerical integration scheme can also be adapted to other problems of interest where the convolution integral with discontinuous density is required to be computed.