Radial perfectly matched layers and infinite elements for the anisotropic wave equation

Abstract

It is well known [1] that anisotropic materials can lead to instabilities for perfectly matched layer (PML) methods, and in particular a successful modification of PMLs to treat general anisotropic elastodynamic problems is still an open problem. To study this question we consider radial PMLs for the scalar anisotropic wave equation. We discuss the origin of the instabilities of convenient PMLs, which can be traced back to an (additional) essential spectrum of the Laplace transformed problem. Following [2] we show that a suitable complex frequency shifted PML scaling removes the former troublesome spectrum. However, this approach does not permit to increase the damping constant and we are left without a meaningful mechanism to decrease the truncation error. As a remedy we apply truncation free approximations such as Hardy space infinite elements and certain “exact” PML methods. We report computational studies confirming the stability of the new numerical methods.