[Une limitation de la reconstruction hydrostatique pour la résolution du système de Saint-Venant]
De par leur capacité à préserver les états dʼéquilibre, les schémas équilibres connaissent actuellement un fort développement dans la résolution des équations de Saint-Venant. En particulier, la reconstruction hydrostatique proposée dans Audusse et al. (2004) [1], couplée à un flux numérique positif, permet de garantir certaines propriétés comme la positivité de la hauteur dʼeau et, donc, dʼéviter certaines instabilités pour traiter les zones sèches. Dans cette note, nous montrons que cette méthode présente un défaut pour certaines combinaisons de pente, taille de maillage et hauteur dʼeau.
Because of their capability to preserve steady states, well-balanced schemes for Shallow Water equations are becoming popular. Among them, the hydrostatic reconstruction proposed in Audusse et al. (2004) [1], coupled with a positive numerical flux, allows to verify important mathematical and physical properties like the positivity of the water height and, thus, to avoid instabilities when dealing with dry zones. In this note, we prove that this method exhibits an abnormal behavior for some combinations of slope, mesh size and water height.
Accepté le :
Publié le :
Olivier Delestre 1, 2 ; Stéphane Cordier 1 ; Frédéric Darboux 3 ; François James 1
@article{CRMATH_2012__350_13-14_677_0,
author = {Olivier Delestre and St\'ephane Cordier and Fr\'ed\'eric Darboux and Fran\c{c}ois James},
title = {A limitation of the hydrostatic reconstruction technique for {Shallow} {Water} equations},
journal = {Comptes Rendus. Math\'ematique},
pages = {677--681},
publisher = {Elsevier},
volume = {350},
number = {13-14},
year = {2012},
doi = {10.1016/j.crma.2012.08.004},
language = {en},
}
TY - JOUR AU - Olivier Delestre AU - Stéphane Cordier AU - Frédéric Darboux AU - François James TI - A limitation of the hydrostatic reconstruction technique for Shallow Water equations JO - Comptes Rendus. Mathématique PY - 2012 SP - 677 EP - 681 VL - 350 IS - 13-14 PB - Elsevier DO - 10.1016/j.crma.2012.08.004 LA - en ID - CRMATH_2012__350_13-14_677_0 ER -
%0 Journal Article %A Olivier Delestre %A Stéphane Cordier %A Frédéric Darboux %A François James %T A limitation of the hydrostatic reconstruction technique for Shallow Water equations %J Comptes Rendus. Mathématique %D 2012 %P 677-681 %V 350 %N 13-14 %I Elsevier %R 10.1016/j.crma.2012.08.004 %G en %F CRMATH_2012__350_13-14_677_0
Olivier Delestre; Stéphane Cordier; Frédéric Darboux; François James. A limitation of the hydrostatic reconstruction technique for Shallow Water equations. Comptes Rendus. Mathématique, Volume 350 (2012) no. 13-14, pp. 677-681. doi : 10.1016/j.crma.2012.08.004. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.08.004/
[1] A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows, SIAM J. Sci. Comput., Volume 25 (2004) no. 6, pp. 2050-2065
[2] Upwind methods for hyperbolic conservation laws with source terms, Comput. & Fluids, Volume 23 (1994), pp. 1049-1071
[3] Efficient well-balanced hydrostatic upwind schemes for shallow-water equations, J. Comput. Phys., Volume 231 (2012) no. 15, pp. 4993-5015
[4] Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws, and Well-Balanced Schemes for Sources, Frontiers in Mathematics, Birkhäuser, 2004
[5] A subsonic-well-balanced reconstruction scheme for shallow water flows, SIAM J. Numer. Anal., Volume 48 (2010) no. 5, pp. 1733-1758
[6] Well-balanced numerical schemes based on a generalized hydrostatic reconstruction technique, Math. Models Methods Appl. Sci., Volume 17 (2007) no. 12, pp. 2055-2113
[7] O. Delestre, Simulation du ruissellement dʼeau de pluie sur des surfaces agricoles (Rain water overland flow on agricultural fields simulation), PhD thesis, Université dʼOrléans, 2010 (in French), tel.archives-ouvertes.fr/INSMI/tel-00531377/fr.
[8] O. Delestre, C. Lucas, P.-A. Ksinant, F. Darboux, C. Laguerre, T.N.T. Vo, F. James, S. Cordier, SWASHES: a library of shallow water analytic solutions for hydraulic and environmental studies, Int. J. Numer. Methods Fluids (July 2012), in press.
[9] A well-balanced scheme for the numerical processing of source terms in hyperbolic equation, SIAM J. Numer. Anal., Volume 33 (1996), pp. 1-16
[10] Numerical resolution of well-balanced shallow water equations with complex source terms, Adv. Water Resour., Volume 32 (2009), pp. 873-884
[11] Quadtree-adaptive tsunami modelling, Ocean Dynam., Volume 61 (2011) no. 9, pp. 1261-1285
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