Section: New Results

Ocean modeling

Participants : Fabien Marche, Antoine Rousseau.

A first discrete formulation for Green-Naghdi equations on unstructured general meshes

We introduce in [17] the first numerical method available in the literature to approximate the solutions of the Green-Naghdi equations on fairly general unstructured meshes. The method relies on coupled elliptic and hyperbolic problems, the first one accounting for a dispersive correction of the free surface flow description provided by the second one, and on discontinuous polynomial approximations of arbitrary order and the construction of discrete differential operators suitable for such non-conforming approximations. It allows to handle general meshes and nonconforming interfaces. A nonlinear stability result is proved, together with the preservation at the discrete level of motionless steady states. Several test cases highlight the accuracy of this discrete formulation.

Quasi-hydrostatic ocean models

In [9], we work on nontraditional models where the so-called traditional approximation on the Coriolis force is removed. In the derivation of the quasi-geostrophic equations, we obtain new terms in δ/ε, where the domain aspect ratio and the Rossby number are both small numbers. We provide here some rigorous crossed-asymptotics with regards to these parameters , prove some mathematical and physical results on the nontraditional models, and situate them among traditional ones. This was also published as lecture notes given by Antoine ROUSSEAU in 2014: see [8].

Interface conditions for ocean models

In [4] we are interested in the search of interface conditions to couple hydrostatic and nonhydrostatic ocean models. To this aim, we consider simplified systems and use a time discretization to handle linear equations. We recall the links between the two models (with the particular role of the aspect ratio $\delta =H/L$) and introduce an iterative method based on the Schwarz algorithm (widely used in domain decomposition methods). The convergence of this method depends strongly on the choice of interface conditions: this is why we look for exact absorbing conditions and their approximations in order to provide tractable and efficient coupling algorithms.

In [3] we present a study of optimized Schwarz domain decomposition methods for Navier-Stokes equations. Once discretized in time, optimal transparent boundary conditions are derived for the resulting Stokes equations, and a series of local approximations for these nonlocal conditions are proposed. Their convergence properties are studied, and numerical simulations are conducted on the test case of the driven cavity. It is shown that conditions involving one or two degrees of freedom can improve the convergence properties of the original algorithm.