Convolution quadrature for the wave equation with a nonlinear impedance boundary condition

TL;DR

This paper develops a convolution quadrature method for solving the wave equation with nonlinear impedance boundary conditions, demonstrating convergence and optimal rates under certain regularity assumptions, supported by numerical experiments.

Contribution

It introduces a boundary integral formulation for nonlinear impedance problems and proves convergence of a fully discrete Galerkin and convolution quadrature scheme without smoothness assumptions.

Findings

Convergence of the discretization is proven without smoothness assumptions.

Optimal convergence rates are achieved when solutions are sufficiently regular.

Numerical experiments in 3D validate the theoretical results.

Abstract

A rarely exploited advantage of time-domain boundary integral equations compared to their frequency counterparts is that they can be used to treat certain nonlinear problems. In this work we investigate the scattering of acoustic waves by a bounded obstacle with a nonlinear impedance boundary condition. We describe a boundary integral formulation of the problem and prove without any smoothness assumptions on the solution the convergence of a full discretization: Galerkin in space and convolution quadrature in time. If the solution is sufficiently regular, we prove that the discrete method converges at optimal rates. Numerical evidence in 3D supports the theory.