Implementing the perfectly matched layer absorbing boundary condition with mimetic differencing schemes

Abstract

This paper concerns the implementation of the perfectly matched layer (PML) absorbing boundary condition in the framework of a mimetic differencing scheme for Maxwell's Equations. We use mimetic versions of the discrete curl operator on irregular logically rectangular grids to implement anisotropic tensor formulation of the PML. The form of the tensor we use is fixed with respect to the grid and is known to be perfectly matched in the continuous limit for orthogonal coordinate systems in which the metric is constant, i.e. Cartesian coordinates, and quasi-perfectly matched (quasi-PML) for curvilinear coordinates. Examples illustrating the effectiveness and long-term stability of the methods are presented. These examples demonstrate that the grid-based coordinate implementation of the PML is effective on Cartesian grids, but generates systematic reflections on grids which are orthogonal but non-Cartesian (quasi-PML). On non-orthogonal grids progressively worse performance of the PML is demonstrated. The paper begins with a summary derivation of the anisotropic formulation of the perfectly matched layer and mimetic differencing schemes for irregular logically rectangular grids.