The Time Domain Lippmann-Schwinger Equation and Convolution Quadrature

TL;DR

This paper introduces a new approach to solving time domain acoustic scattering problems using a volume Lippmann-Schwinger integral equation, convolution quadrature, and trigonometric collocation, with proven convergence and promising numerical results.

Contribution

It develops a novel method combining convolution quadrature and collocation for the time domain Lippmann-Schwinger equation, with theoretical analysis and initial numerical validation.

Findings

Unique solution existence for the equation.

Conditional convergence and error estimates established.

Numerical results indicate robustness even with discontinuous sound speeds.

Abstract

We consider time domain acoustic scattering from a penetrable medium with a variable sound speed. This problem can be reduced to solving a time domain volume Lippmann-Schwinger integral equation. Using convolution quadrature in time and trigonometric collocation in space we can compute an approximate solution. We prove that the time domain Lippmann-Schwinger equation has a unique solution and prove conditional convergence and error estimates for the fully discrete solution for smooth sound speeds. Preliminary numerical results show that the method behaves well even for discontinuous sound speeds.