Long-Time Stability and Convergence of the Uniaxial Perfectly Matched Layer Method for Time-Domain Acoustic Scattering Problems

Abstract

The uniaxial perfectly matched layer (PML) method uses a rectangular domain to define the PML problem and thus provides greater flexibility and efficiency in dealing with problems involving anisotropic scatterers. In this paper we first derive the uniaxial PML method for solving the time-domain scattering problem based on the Laplace transform and complex coordinate stretching in the frequency domain. We prove the long-time stability of the initial-boundary value problem of the uniaxial PML system for piecewise constant medium properties and show the exponential convergence of the time-domain uniaxial PML method. Our analysis shows that for fixed PML absorbing medium properties, any error of the time-domain PML method can be achieved by enlarging the thickness of the PML layer as $\ln T$ for large $T>0$. Numerical experiments are included to illustrate the efficiency of the PML method.