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

Nora  Aïssiouene; Marie-Odile  Bristeau; Edwige  Godlewski; Jacques  Sainte-Marie; Nora  Aïssiouene; Marie-Odile  Bristeau; Edwige  Godlewski; Jacques  Sainte-Marie

doi:10.3934/nhm.2016.11.1

Networks and Heterogeneous Media

2016, Volume 11, Issue 1: 1-27. doi: 10.3934/nhm.2016.11.1

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

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

Received: 01 June 2015 Revised: 01 October 2015
Primary: 65M06, 65M60, 76M10; Secondary: 76B15.

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.
- Projection method,
- non-hydrostatic,
- Navier-Stokes,
- Euler,
- free surface,
- depth-averaged Euler system,
- dispersive
Citation: Nora Aïssiouene, Marie-Odile Bristeau, Edwige Godlewski, Jacques Sainte-Marie. A combined finite volume - finite element scheme for a dispersive shallow water system[J]. Networks and Heterogeneous Media, 2016, 11(1): 1-27. doi: 10.3934/nhm.2016.11.1

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Abstract

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.

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