The role of numerical boundary procedures in the stability of perfectly matched layers

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 variable coefficient PML. 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 result

ADER-DG - Analysis, further Development and Applications

After introducing the Discontinuous Galerkin (DG) method a detailed misfit analysis on its numerical approximation is performed. We investigate the accuracy of the scheme, the element type (tetrahedrons and hexahedrons), the spatial sampling of the computational domain and the number of propagated wavelengths. As the error norm we chose a time-frequency representation, which illustrates the time evolution of the spectral content. The results of this analysis are confirmed by a multi-institutional code validation project. In order to improve efficiency, we expand the computer code to non-conforming, hybrid meshes. In 2 dimensions, riangulars and quadrilaterals can be combined within one computational domain. Several convergence tests are carried out and the newly invented scheme is applied to different test cases including thin layers and variable material. Furthermore, as absorbing boundaries suffer from spurious reflections at artificial boundaries of the computational domain, we introduce a convolutional perfectly matched layer (CPML) to the scheme. Due to the loss of definite stability, we accomplish several test cases in order to examine the scheme’s behavior. A switchoff criterion for the CPML is suggested. Considering topographic effects on seismic waves, we perform a systematic study of different parameterizations involving the wave type and frequency of the input signal, the dataset resolution and various amplification factors of real topography in the region of Grenoble, France. Special events are simulated at Mt. Hochstaufen, Southern Bavaria, and compared to real recordings