Comptes Rendus
no. 13-14
Numerical Analysis
A consistent intermediate wave speed for a well-balanced HLLC solver
[Une vitesse de propogation intermédiaire consistante pour un schéma HLLC bien équilibré]

Présenté par : Philippe G. Ciarlet

Enrique D. Fernández-Nieto1 ; Didier Bresch2 ; Jérôme Monnier3
1 Dpto Matemática Aplicada I, Universidad de Sevilla, Avda. Reina Mercedes N. 2, 41012 Sevilla, Spain
2 LAMA, UMR5127 CNRS, Université de Savoie, 73376 Le Bourget du Lac, France
3 LJK – MOISE project-team, INP-Grenoble & INRIA, BP 53, 38041 Grenoble, France
Comptes Rendus. Mathématique, Volume 346 (2008) no. 13-14, pp. 795-800.

Nous présentons une modification du schéma HLLC pour application aux équations de Saint-Venant non homogènes avec transport de polluants. Cette nouvelle version du schéma est reliée à la définition d'une vitesse d'onde intermédiaire consistante. Un exemple est donné afin d'illustrer les conséquences d'une mauvaise approximation de la vitesse intermédiaire sur des résultats de l'approximation de la concentration en polluant même avec un schéma bien équilibré.

In this work, we present a HLLC scheme modification for application to non-homogeneous shallow-water equations with pollutant transport. This new version is related to the definition of a consistent approximation of the intermediate wave speed. Numerical results are presented to illustrate the importance of such approximation to get appropriate pollutant concentration profiles.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2008.05.012
Enrique D. Fernández-Nieto 1 ; Didier Bresch 2 ; Jérôme Monnier 3

1 Dpto Matemática Aplicada I, Universidad de Sevilla, Avda. Reina Mercedes N. 2, 41012 Sevilla, Spain
2 LAMA, UMR5127 CNRS, Université de Savoie, 73376 Le Bourget du Lac, France
3 LJK – MOISE project-team, INP-Grenoble & INRIA, BP 53, 38041 Grenoble, France 
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     title = {A consistent intermediate wave speed for a well-balanced {HLLC} solver},
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Enrique D. Fernández-Nieto; Didier Bresch; Jérôme Monnier. A consistent intermediate wave speed for a well-balanced HLLC solver. Comptes Rendus. Mathématique, Volume 346 (2008) no. 13-14, pp. 795-800. doi : 10.1016/j.crma.2008.05.012. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2008.05.012/

[1] E. Audusse; F. Bouchut; M.O. Bristeau; R. Klein; B. Perthame A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows, SIAM J. Sci. Comp., Volume 25 (2004) no. 6, pp. 2050-2065

[2] A. Bermúdez; M.E. Vázquez Cendón Upwind methods for hyperbolic conservation laws with source terms, Computers Fluids, Volume 23-8 (1994), pp. 1049-1071

[3] F. Bouchut Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws and Well-Balanced Schemes for Sources, Birkhäuser, 2004

[4] T. Chacón; A. Domínguez; E.D. Fernández-Nieto A family of stable numerical solvers for the shallow water equations with source terms, Comput. Methods Appl. Mech. Engrg., Volume 192 (2003), pp. 203-225

[5] T. Chacón Rebollo; A. Domínguez Delgado; E.D. Fernández-Nieto Asymptotically balanced schemes for non-homogeneous hyperbolic systems – application to shallow water equations, C. R. Acad. Sci. Paris, Ser. I, Volume 338 (2004), pp. 85-90

[6] L. Gosse A well-balanced scheme using non-conservative products designed for hyperbolic system of conservation laws with source terms, Math. Models Methods Appl. Sci., Volume 11 (2001), pp. 339-365

[7] G. Dal Maso; P.G. LeFloch; F. Murat Definition and weak stability of nonconservative products, J. Math. Pures Appl., Volume 74 (1995), pp. 483-548

[8] A. Domínguez, E.D. Fernández-Nieto, S. Martín, A well balanced third order HLL scheme for hyperbolic systems with source term, Application to shallow water equations (2006), submitted for publication

[9] E.D. Fernández-Nieto, G. Narbona-Reina, Extension of WAF type method to non-homogeneous shallow water equations with pollutant, J. Sci. Comput. (2008), in press

[10] J.M. Greenberg; A.Y. Leroux A well-balanced scheme for the numerical processing of source terms in hyperbolic equations, SIAM J. Numer. Anal., Volume 33 (1996) no. 1, pp. 1-16

[11] P.G. LeFloch; A.E. Tzavaras Representation and weak limits and definition of nonconservative products, SIAM J. Math. Anal., Volume 30 (1999), pp. 1309-1342

[12] R.J. LeVeque Balancing source terms and flux gradients in high-resolution Godunov methods: The quasi-steady wave-propagation algorithm, J. Comput. Phys., Volume 146 (1998), pp. 346-365

[13] C. Parés; M.J. Castro On the well-balanced property of Roe method for nonconservative hyperbolic systems. Applications to shallow water systems, ESAIM: M2AN, Volume 38 (2004) no. 5, pp. 821-852

[14] B. Perthame; C. Simeoni A kinetic scheme for the Saint-Venant system with a source term, Calcolo, Volume 38 (2001) no. 4, pp. 201-231

[15] B. Perthame; C. Simeoni Convergence of the upwind interface source method for hyperbolic conservation laws (T. Hou; E. Tadmor, eds.), Proceeding of Hyp 2002, Springer, 2003

[16] E.F. Toro Shock-Capturing Methods for Free-Surface Shallow Flows, Springer, 1999

[17] J.G. Zhou; D.M. Causon; C.G. Mingham; D.M. Ingram The surface gradient method for the treatment of source terms in the sallow-water equations, J. Comput. Phys., Volume 168 (2001), pp. 1-25

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