## Section: New Results

Keywords : Perfectly Matched Layer, Smith factorization, Water Waves.

### Perfectly Matched Layers for the Shallow Water equations

Participants : Hélène Barucq, Julien Diaz, Mounir Tlemcani [ Assistant Professor, University of Oran, Algeria ] .

The Perfectly Matched Layer (PML) technique for the numerical absorption of waves, initially introduced about 20 years ago by Bérenger [48] in electromagnetism, is now widely used for simulating the propagation of waves in unbounded domains, in particular in time domain acoustics. However, this technique induces strong instabilities when applied to Euler equations [60] or to shallow water equations. Much works have been devoted to the stabilization of the PML for linearized Euler equations [59] , [64] and we propose here a stable PML for the linearized shallow water equations with a Coriolis term and a uniform mean flow. The technique follows the one proposed by Nataf [64] for linearized Euler equations and rely on the use of the Smith factorization, which allows us to split the vorticity waves from the advective and entropy waves and to treat these three types of waves separately. Indeed the propagation of vorticity and entropy waves is governed by a classical transport equation and these waves can be easily absorbed by a transparent condition at the end of the layer. To absorb the advective waves, we first have to use the transformation proposed by Hu [59] before applying a classical PML technique to avoid an exponential growth of the waves.