We are interested in the modeling and the numerical approximation of flows in the presence of a roof, for example flows in sewers or under an ice floe. A shallow water model with a supplementary congestion constraint describing the roof is derived from the Navier-Stokes equations. The congestion constraint is a challenging problem for the numerical resolution of hyperbolic equations. To overcome this difficulty, we follow a pseudo-compressibility relaxation approach. Eventually, a numerical scheme based on a finite volume method is proposed. The well-balanced property and the dissipation of the mechanical energy, acting as a mathematical entropy, are ensured under a non-restrictive condition on the time step in spite of the large celerity of the potential waves in the congested areas. Simulations in one dimension for transcritical steady flow are carried out and numerical solutions are compared to several analytical (stationary and non-stationary) solutions for validation.
Mots-clés : Shallow water equations, congested hyperbolic model, unilateral constraint, well-balanced scheme, entropic scheme
@article{M2AN_2018__52_5_1679_0,
author = {Godlewski, Edwige and Parisot, Martin and Sainte-Marie, Jacques and Wahl, Fabien},
title = {Congested shallow water model: roof modeling in free surface flow},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1679--1707},
publisher = {EDP-Sciences},
volume = {52},
number = {5},
year = {2018},
doi = {10.1051/m2an/2018032},
zbl = {1417.35121},
mrnumber = {3878609},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an/2018032/}
}
TY - JOUR AU - Godlewski, Edwige AU - Parisot, Martin AU - Sainte-Marie, Jacques AU - Wahl, Fabien TI - Congested shallow water model: roof modeling in free surface flow JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2018 SP - 1679 EP - 1707 VL - 52 IS - 5 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an/2018032/ DO - 10.1051/m2an/2018032 LA - en ID - M2AN_2018__52_5_1679_0 ER -
%0 Journal Article %A Godlewski, Edwige %A Parisot, Martin %A Sainte-Marie, Jacques %A Wahl, Fabien %T Congested shallow water model: roof modeling in free surface flow %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2018 %P 1679-1707 %V 52 %N 5 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an/2018032/ %R 10.1051/m2an/2018032 %G en %F M2AN_2018__52_5_1679_0
Godlewski, Edwige; Parisot, Martin; Sainte-Marie, Jacques; Wahl, Fabien. Congested shallow water model: roof modeling in free surface flow. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 5, pp. 1679-1707. doi : 10.1051/m2an/2018032. http://archive.numdam.org/articles/10.1051/m2an/2018032/
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