Lacunae based stabilization of PMLs

Perfectly matched layers (PMLs) are used for the numerical solution of wave propagation problems on unbounded regions. They surround the finite computational domain (obtained by truncation) and are designed to attenuate and completely absorb all the outgoing waves while producing no reflections from the interface between the domain and the layer. PMLs have demonstrated excellent performance for many applications. However, they have also been found prone to instabilities that manifest themselves when the simulation time is long. Hereafter, we propose a modification that stabilizes any PML applied to a hyperbolic partial differential equation/system that satisfies the Huygens’ principle (such as the 3D d’Alembert equation or Maxwell’s equations in vacuum). The modification makes use of the presence of lacunae in the corresponding solutions and allows us to establish a temporally uniform error bound for arbitrarily long-time intervals. At the same time, it does not change the original PML equations. Hence, the matching properties of the layer, as well as any other properties deemed important, are fully preserved. We also emphasize that besides the aforementioned PML instabilities per se, the methodology can be used to cure any other undesirable long-term computational phenomenon, such as the accuracy loss of low order absorbing boundary conditions.