The Role of Numerical Boundary Procedures in the Stability of Perfectly Matched Layers

Abstract

In this paper, we address the temporal energy growth associated with numerical approximations of the perfectly matched layer (PML) for Maxwell's equations in first order form. In the literature, several studies have shown that a numerical method which is stable in the absence of the PML can become unstable when the PML is introduced. We demonstrate in this paper that this instability can be directly related to numerical treatment of boundary conditions in the PML. First, at the continuous level, we establish the stability of the constant coefficient initial boundary value problem for the PML. To enable the construction of stable numerical boundary procedures, we derive energy estimates for the PML in the Laplace space. Second, we develop a high order accurate and stable numerical approximation for the PML using summation-by-parts finite difference operators to approximate spatial derivatives and weak enforcement of boundary conditions using penalties. By constructing analogous discrete energy estimates we show discrete stability and convergence of the numerical method. Numerical experiments verify the theoretical results.