Mathematical Problems in Mechanics/Partial Differential Equations
A new model of Saint Venant and Savage–Hutter type for gravity driven shallow water flows
[Un nouveau modèle de type Saint Venant et Savage–Hutter pour les écoulements gravitaires en eaux peu profondes]
[Un nouveau modèle de type Saint Venant et Savage–Hutter pour les écoulements gravitaires en eaux peu profondes]
Comptes Rendus. Mathématique, Volume 336 (2003) no. 6, pp. 531-536.
Nous introduisons un nouveau modèle pour les écoulements en eaux peu profondes avec fond variable. Un prototype est l'équation de Saint Venant pour les rivières et zones côtières, qui est valable pour des pentes faibles. Un modèle amélioré, dû à Savage–Hutter, est valable pour de petites variations de pente. Nous introduisons un nouveau modèle valable quelle que soit la topographie, et qui a les propriétés (i) de fournir une inégalité de dissipation d'énergie, (ii) d'être une solution hydrostatique exacte des équations d'Euler. La difficulté que nous résolvons est la dépendance du champ de vitesse dans la variable normale au fond, que nous sommes capables d'établir exactement. Les applications visées sont le calcul numérique des écoulements granulaires (par exemple avalanches de débris) pour lequel un tel modèle est particulièrement adapté.
We introduce a new model for shallow water flows with non-flat bottom. A prototype is the Saint Venant equation for rivers and coastal areas, which is valid for small slopes. An improved model, due to Savage–Hutter, is valid for small slope variations. We introduce a new model which relaxes all restrictions on the topography. Moreover it satisfies the properties (i) to provide an energy dissipation inequality, (ii) to be an exact hydrostatic solution of Euler equations. The difficulty we overcome here is the normal dependence of the velocity field, that we are able to establish exactly. Applications we have in mind concern, in particular, computational aspects of flows of granular material (for example in debris avalanches) where such models are especially relevant.
Reçu le :
Révisé le :
Publié le :
DOI :
10.1016/S1631-073X(03)00117-1
Révisé le :
Publié le :
Affiliations des auteurs :
François Bouchut 1 ; Anne Mangeney-Castelnau 2 ; Benoı̂t Perthame 1, 3 ; Jean-Pierre Vilotte 2
@article{CRMATH_2003__336_6_531_0,
author = {Fran\c{c}ois Bouchut and Anne Mangeney-Castelnau and Beno{\i}̂t Perthame and Jean-Pierre Vilotte},
title = {A new model of {Saint} {Venant} and {Savage{\textendash}Hutter} type for gravity driven shallow water flows},
journal = {Comptes Rendus. Math\'ematique},
pages = {531--536},
publisher = {Elsevier},
volume = {336},
number = {6},
year = {2003},
doi = {10.1016/S1631-073X(03)00117-1},
language = {en},
}
TY - JOUR AU - François Bouchut AU - Anne Mangeney-Castelnau AU - Benoı̂t Perthame AU - Jean-Pierre Vilotte TI - A new model of Saint Venant and Savage–Hutter type for gravity driven shallow water flows JO - Comptes Rendus. Mathématique PY - 2003 SP - 531 EP - 536 VL - 336 IS - 6 PB - Elsevier DO - 10.1016/S1631-073X(03)00117-1 LA - en ID - CRMATH_2003__336_6_531_0 ER -
%0 Journal Article %A François Bouchut %A Anne Mangeney-Castelnau %A Benoı̂t Perthame %A Jean-Pierre Vilotte %T A new model of Saint Venant and Savage–Hutter type for gravity driven shallow water flows %J Comptes Rendus. Mathématique %D 2003 %P 531-536 %V 336 %N 6 %I Elsevier %R 10.1016/S1631-073X(03)00117-1 %G en %F CRMATH_2003__336_6_531_0
François Bouchut; Anne Mangeney-Castelnau; Benoı̂t Perthame; Jean-Pierre Vilotte. A new model of Saint Venant and Savage–Hutter type for gravity driven shallow water flows. Comptes Rendus. Mathématique, Volume 336 (2003) no. 6, pp. 531-536. doi : 10.1016/S1631-073X(03)00117-1. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(03)00117-1/
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- A Saint-Venant model for gravity driven shallow water flows with variable density and compressibility effects, Mathematical and Computer Modelling, Volume 47 (2008) no. 3-4, pp. 436-444 | DOI:10.1016/j.mcm.2007.04.016 | Zbl:1130.76021
- Shock waves in rapid flows of dense granular materials: Theoretical predictions and experimental results, Physical Review E, Volume 78 (2008) no. 4 | DOI:10.1103/physreve.78.041308
- An exact solution for ideal dam‐break floods on steep slopes, Water Resources Research, Volume 44 (2008) no. 1 | DOI:10.1029/2007wr006353
- Recent mathematical results and open problems about shallow water equations, Analysis and simulation of fluid dynamics. Collected papers based on the presentations at the conference, Lille, France, June 2005, Basel: Birkhäuser, 2007, pp. 15-31 | DOI:10.1007/978-3-7643-7742-7_2 | Zbl:1291.35001
- The Navier-Stokes and Euler Equations — Fluid and Gas Dynamics, Applied Partial Differential Equations (2007), p. 21 | DOI:10.1007/978-3-540-34646-3_3
- Derivation of a new two-dimensional viscous shallow water model with varying topography, bottom friction and capillary effects, European Journal of Mechanics. B. Fluids, Volume 26 (2007) no. 1, pp. 49-63 | DOI:10.1016/j.euromechflu.2006.04.007 | Zbl:1105.76021
- Plasticity and geophysical flows: A review, Journal of Non-Newtonian Fluid Mechanics, Volume 142 (2007) no. 1-3, p. 4 | DOI:10.1016/j.jnnfm.2006.05.005
- Morphodynamic modeling of erodible laminar channels, Physical Review E, Volume 76 (2007) no. 5 | DOI:10.1103/physreve.76.056318
- Dam-break flow for arbitrary slopes of the bottom, Journal of Engineering Mathematics, Volume 54 (2006) no. 4, pp. 319-331 | DOI:10.1007/s10665-006-9034-5 | Zbl:1095.76011
- A well-balanced positivity preserving “second-order” scheme for shallow water flows on unstructured meshes, Journal of Computational Physics, Volume 206 (2005) no. 1, pp. 311-333 | DOI:10.1016/j.jcp.2004.12.016 | Zbl:1087.76072
- On measure-valued solutions to a two-dimensional gravity-driven avalanche flow model, Mathematical Methods in the Applied Sciences, Volume 28 (2005) no. 18, pp. 2201-2223 | DOI:10.1002/mma.660 | Zbl:1090.35118
- Multivalued equations for granular avalanches, Nonlinear Analysis. Theory, Methods Applications. Series A: Theory and Methods, Volume 62 (2005) no. 5, pp. 895-912 | DOI:10.1016/j.na.2005.03.107 | Zbl:1074.35070
- Wave patterns for shallow water equations, Quarterly of Applied Mathematics, Volume 63 (2005) no. 2, pp. 225-249 | DOI:10.1090/s0033-569x-05-00939-6 | Zbl:1082.76050
- A Fast and Stable Well-Balanced Scheme with Hydrostatic Reconstruction for Shallow Water Flows, SIAM Journal on Scientific Computing, Volume 25 (2004) no. 6, p. 2050 | DOI:10.1137/s1064827503431090
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