High-order boundary integral equation solution of high frequency wave scattering from obstacles in an unbounded linearly stratified medium

TL;DR

This paper introduces a boundary integral equation method for high-frequency wave scattering in a linearly stratified medium, achieving high accuracy and efficiency for large obstacles, with broad applications in physics and engineering.

Contribution

It is the first application of boundary integral equations to 2D wave scattering in a stratified medium with efficient fundamental solution evaluation.

Findings

Achieved 11-digit accuracy for obstacles 50 wavelengths across

Efficient fundamental solution evaluation with frequency-independent effort

Solved high-frequency scattering problems rapidly on standard hardware

Abstract

We apply boundary integral equations for the first time to the two-dimensional scattering of time-harmonic waves from a smooth obstacle embedded in a continuously-graded unbounded medium. In the case we solve the square of the wavenumber (refractive index) varies linearly in one coordinate, i.e. $(\Delta + E + x_2)u(x_1,x_2) = 0$ where $E$ is a constant; this models quantum particles of fixed energy in a uniform gravitational field, and has broader applications to stratified media in acoustics, optics and seismology. We evaluate the fundamental solution efficiently with exponential accuracy via numerical saddle-point integration, using the truncated trapezoid rule with typically 100 nodes, with an effort that is independent of the frequency parameter $E$. By combining with high-order Nystrom quadrature, we are able to solve the scattering from obstacles 50 wavelengths across to 11 digits of accuracy in under a minute on a desktop or laptop.