Discrete stability of perfectly matched layers for anisotropic wave equations in first and second order formulation

Analysis of standard central finite difference discretization of perfectly matched layers for a scalar wave equation violating the geometric stability condition is presented. Both first and second order formulations are considered, and in the continuous setting our layers support temporally growing modes. The analysis shows that the discrete second order layer has much better stability properties than the discrete first order layer. In particular the second order layer exhibits growth only if well resolved growing modes are represented on the grid, while the first order version is unstable at most resolutions. We generalize this instability result to other first order systems.