Convolution quadrature for the wave equation with a nonlinear impedance boundary condition
Provides the first rigorous convergence analysis for time-domain boundary integral methods applied to nonlinear impedance boundary conditions, addressing a gap in numerical analysis for wave propagation problems.
The authors develop and analyze a boundary integral formulation for acoustic wave scattering with a nonlinear impedance boundary condition, proving convergence of a Galerkin-convolution quadrature discretization without smoothness assumptions and achieving optimal rates for regular solutions.
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.