# American Institute of Mathematical Sciences

March  2016, 11(1): 1-27. doi: 10.3934/nhm.2016.11.1

## A combined finite volume - finite element scheme for a dispersive shallow water system

 1 Inria, EPC ANGE, 2 rue Simone Iff, F75012 Paris, France, France, France, France

Received  June 2015 Revised  October 2015 Published  January 2016

We propose a variational framework for the resolution of a non-hydrostatic Saint-Venant type model with bottom topography. This model is a shallow water type approximation of the incompressible Euler system with free surface and slightly differs from the Green-Nagdhi model, see [13] for more details about the model derivation.
The numerical approximation relies on a prediction-correction type scheme initially introduced by Chorin-Temam [17] to treat the incompressibility in the Navier-Stokes equations. The hyperbolic part of the system is approximated using a kinetic finite volume solver and the correction step implies to solve a mixed problem where the velocity and the pressure are defined in compatible finite element spaces.
The resolution of the incompressibility constraint leads to an elliptic problem involving the non-hydrostatic part of the pressure. This step uses a variational formulation of a shallow water version of the incompressibility condition.
Several numerical experiments are performed to confirm the relevance of our approach.
Citation: Nora Aïssiouene, Marie-Odile Bristeau, Edwige Godlewski, Jacques Sainte-Marie. A combined finite volume - finite element scheme for a dispersive shallow water system. Networks & Heterogeneous Media, 2016, 11 (1) : 1-27. doi: 10.3934/nhm.2016.11.1
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##### References:
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