Numerical simulation of a depth-averaged Euler
system and comparison with analytical solutions
N. Aı̈ssiouene, M-O. Bristeau, E. Godlewski, J. Sainte-Marie
May 22, 2014
Abstract
We propose an efficient numerical scheme for the resolution of a nonhydrostatic Saint-Venant type model. The model with dispersive effects
is a shallow water type approximation of the incompressbile Euler system
with free surface and slightly differs from the Green-Naghdi model, (see [2]
for details about the model derivation).
The numerical approximation relies on a projection-correction type
scheme [3]. The hyperbolic part of the system is approximated using a
kinetic based finite volume solver [1] and the correction step implies an
elliptic problem involving the non-hydrostatic part of the pressure. The
numerical scheme satisfies classical properties (positivity, well-balancing)
and is confronted with various analytical solutions (stationary and timedependent). Notice that the numerical procedure remains stable when the
water depth tends to zero.
References
[1] Emmanuel Audusse, Marie-Odile Bristeau, and Benoit Perthame. Kinetic
Schemes for Saint-Venant Equations with Source Terms on Unstructured
Grids. Rapport de recherche RR-3989, INRIA, 2000. Projet M3N.
[2] M. O. Bristeau, A. Mangeney, J. Sainte-Marie, and N. Seguin. An energyconsistent depth-averaged euler system: derivation and properties. submitted, 2014.
[3] A. J. Chorin. Numerical solution of the Navier-Stokes equations. Math.
Comp., 22:745–762, 1968.
1