Penalty Sponge Layers (PSL) for hyperbolic systems. General formulation, well-posedness and stability

Abstract

In this work we design and analyze an alternative to the Absorbing Boundary Conditions (ABC) or Perfectly Matched Layers (PML) technique to solve symmetric first order hyperbolic systems in unbounded domains. We propose a general formulation of absorbing layers that we call Penalty Sponge Layer (PSL). Well-posedness as well as stability properties are quite simply inherited from the original problem. Such a formulation is written in terms of a set of auxiliary variables similar to those encountered in the Complex Frequency Shifted PML (CFS-PML) technique, but this time they have the role of a penalty term, hence the name PSL. This second role of penalty, ensured by stretching an artificial frequency to infinity in the vicinity of the interface, makes the layer asymptotically non-reflecting. Numerical tests are performed to compare them with Standard Sponge Layers (SSL) and PMLs, and to validate their success, even in severe situations where the latter fail (e.g., the Euler system at oblique subsonic/critical or supersonic flow and anisotropic waves).