Abstract
In the current context of global warming, the development of new strategies for preventing risks of coastal submersion associated with storms becomes crucial. Such strategies should rely on accurate numerical simulations. To achieve this, it is necessary on one hand to work with suitable mathematical models, with a good description of all the physical processes at stake, and on the other hand to design new numerical methods able to capture the particular assets of the models. Hence, the goal of this work is to contribute to the development of new numerical tools dedicated to the approximations of several shallow water asymptotics with weakly dispersive and fully nonlinear effects: namely the Green-naghdi equations and the rotational Green-Naghdi equations which include new terms able to account for nonlinear wave-current interactions. We develop two new discrete formulations belonging to the family of discontinuous Galerkin schemes: one for the elliptic problem associated with the Green-Naghdi dispersive terms, which is proved to be coercive, leading to an invertible matrix, and one for the rotational Green-Naghdi equations, ensuring the positivity of the water height and the enstrophy.