Section: New Results
Accurate simulations of fluid flows
A numerical scheme for the Saint-Venant–Exner equations
Participants : Emmanuel Audusse, Philippe Ung.
After having established a Godunov-type method based on the design of a three-wave Approximate Riemann Solver for the Saint-Venant equations  , we extended this approach to the Saint-Venant–Exner equations for modelling the sediment transport. The coupled aspect between the hydraulic and the morphodynamic parts is only located on the evaluation of the wave velocities. Under this assumption, the proposed scheme can be interpreted as a hybrid method between the splitting and non-splitting methods and it also raises the issue of the choice between the two previous approaches.
These results were proven in collaboration with Christophe Chalons from Univ. Versailles–Saint-Quentin.
Simulations of fluid/particules interactions
Participant : Nicolas Seguin.
In collaboration with Nina Aguillon and Frédéric Lagoutière from Univ. Paris-Sud, we proved in  the convergence of finite volume schemes for a simplified model of fluid-particle interaction. The mesh follows the particle which appears in the model as a pointwise contribution. The numerical scheme is based on local well-balanced fluxes, which permits to obtain compactness and convergence.
Participants : Emmanuel Audusse, Marie-Odile Bristeau, Jacques Sainte-Marie.
The hydrostatic reconstruction is a general and efficient method to handle source terms that uses an arbitrary solver for the homogeneous problem and leads to a consistent, well-balanced, positive scheme satisfying a semi-discrete entropy inequality.
In  , we proved with Francois Bouchut from Univ. Marne-la-Vallée that the hydrostatic reconstruction coupled to the classical kinetic solver satisfies a fully discrete entropy inequality which involves an error term but the latter goes to zero strongly with the mesh size.
A numerical scheme for multilayer shallow-water model for all Froude regimes
Participant : Martin Parisot.
The aim of this work in collaboration with Jean-Paul Vila from INSA/IMT is to propose an efficient numerical resolution to simulate stratified non-miscible fluids. The strategy should be consistent for all regime especially with the so-called low-Froude regime particularly relevant for applications. The proposed scheme is entropy-satisfying, well-balanced and asymptotic preserving. In addition the stability of the scheme is ensured for large time scale. More precisely, it does not depend on the gravity waves, which are very restrictive for the targeted applications, such as oceanology and meteorology. Further work using the strategy for sustainable energies is in progress.
Adaptation of the Godunov scheme to the low Froude regime
Participants : Emmanuel Audusse, Do Minh Hieu, Yohan Penel.
Standard numerical schemes designed for the simulation of fluid flows are known to fail when the Mach number becomes too small. Similar behaviours are observed for geophysical flows when the Froude number decreases. Do Minh Hieu is interested in the numerical simulation of the Shallow Water equations including some Coriolis forces. He investigated several corrections of the standard Godunov schemes in 1D to preserve the kernel of spatial operators involved in the aformentioned equations and blamed for being responsible of the loss of accuracy. He now intends to perform the same analysis in 2D under the supervision of E. Audusse, S. Dellacherie (from CEA), P. Omnès (from CEA) and Y. Penel.