Well-posedness and convergence analysis of PML method for time-dependent acoustic scattering problems over a locally rough surface

TL;DR

This paper establishes the well-posedness and stability of a PML method for simulating time-dependent acoustic scattering by a locally rough surface, proving exponential convergence of the PML solution to the true solution.

Contribution

It introduces a novel PML scheme with a specially designed absorbing medium for acoustic scattering over rough surfaces, with rigorous convergence analysis.

Findings

Proved well-posedness and stability of the PML scheme.

Demonstrated exponential convergence of PML solutions.

Numerical results confirm theoretical convergence rates.

Abstract

We aim to analyze and calculate time-dependent acoustic wave scattering by a bounded obstacle and a locally perturbed non-selfintersecting curve. The scattering problem is equivalently reformulated as an initial-boundary value problem of the wave equation in a truncated bounded domain through a well-defined transparent boundary condition. Well-posedness and stability of the reduced problem are established. Numerically, we adopt the perfect matched layer (PML) scheme for simulating the propagation of perturbed waves. By designing a special absorbing medium in a semi-circular PML, we show well-posedness and stability of the truncated initial-boundary value problem. Finally, we prove that the PML solution converges exponentially to the exact solution in the physical domain. Numerical results are reported to verify the exponential convergence with respect to absorbing medium parameters and thickness of the PML.