We are interested in simulating wave propagation in
three-dimensional infinite anisotropic media which can be described with Maxwell's equations. Since the medium is infinite, one needs to introduce an equivalent formulation posed in a bounded domain which is suitable for numerical purposes. The widely-used Perfectly Matched Layer (PML) method consists in surrounding the computational domain by layers which absorb outgoing waves. They can be used in the time and frequency domains. It is well known that classical Cartesian PMLs fail in the presence of backward waves in the PML direction. Although backward waves do not exist for diagonal anisotropy for the 3D scalar wave equation as well as for 2D Maxwell's equations, surprisingly they do exist for 3D Maxwell's equations for a class of diagonal anisotropic dielectric tensors. This is counterintuitive and I will give a more detailed analysis.
In the second part of my presentation I will consider whether time-domain instabilities imply a lack of convergence towards the physical solution in the frequency domain. In order to understand this question we consider acoustic wave propagagtion in anisotropic media. We expect a similar behaviour for Maxwell's equations, however this work is still in progress.