Boundary conditions and entropy stability for nearshore wave models and their numerical discretization

This talk deals with several issues arising in the modelling and
numerical approximation of water waves in the nearshore zone.

In the first part, we focus on boundary conditions for a dispersive
Boussinesq-type model. We provide a rigorous analysis of boundary
conditions that allow for wave generation and absorption without relying
on artificial sponge layers. Based on this analysis, we propose a
second-order accurate numerical scheme and illustrate its effectiveness.

The second part addresses wave breaking through a coupling between the
dispersive model and the hyperbolic shallow water system. Numerical
simulations are presented and compared with laboratory experiments,
showing good agreement.

In the final part, we study a numerical discretization of the shallow
water system based on a kinetic formulation. The resulting scheme is
positive, well-balanced, and satisfies a fully discrete entropy
inequality owing to an implicit time stepping.