Time domain analysis and localization of a non-local PML for dispersive wave equations

Abstract

In this work we design and analyze new perfectly matched layers (PML) for a dispersive waves equation: the Klein Gordon equation. We show that because of the dispersion, classical PMLs do not guarantee the convergence to zero of the error, which hampers the precision in long time simulation. We propose to consider a non-local PML for which we can obtain explicit uniform estimates for the reflected analytical solution in time domain, given by an integral representation formula. This uniform estimates ensure the convergence of the error to zero at fixed time t and guarantee the accuracy of the layer. For the implementation of the new PML, we propose a localization technique that we validate numerically.