Perfectly matched layers in 1-d : energy decay for continuous and semi-discrete waves

In this paper we investigate the efficiency of the method of perfectly matched layers (PML) for the 1-d wave equation. The PML method furnishes a way to compute solutions of the wave equation for exterior problems in a finite computational domain by adding a damping term on the matched layer. In view of the properties of solutions in the whole free space, one expects the energy of solutions obtained by the PML method to tend to zero as t -> infinity, and the rate of decay can be understood as a measure of the efficiency of the method. We prove, indeed, that the exponential decay holds and characterize the exponential decay rate in terms of the parameters and damping potentials entering in the implementation of the PML method. We also consider a space semi-discrete numerical approximation scheme and we prove that, due to the high frequency spurious numerical solutions, the decay rate fails to be uniform as the mesh size parameter h tends to zero. We show however that adding a numerical viscosity term allows us to recover the property of exponential decay of the energy uniformly on h. Although our analysis is restricted to finite differences in 1-d, most of the methods and results apply to finite elements on regular meshes and to multi-dimensional problems.