Comptes Rendus
no. 1
Numerical Analysis
Asymptotically balanced schemes for non-homogeneous hyperbolic systems – application to the Shallow Water equations
[Schémas asymptotiquement equilibrés pour systèmes hyperboliques non-homogénes – application aux équations de Saint-Venant]

Présenté par : Olivier Pironneau

Tomás Chacón Rebollo1 ; Antonio Domı́nguez Delgado2 ; Enrique D. Fernández Nieto2
1 Dpto Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, c/Tarfia s/n, 41012 Sevilla, Spain
2 Dpto Matemática Aplicada I, Universidad de Sevilla, Avda. Reina Mercedes N. 2, 41012 Sevilla, Spain
Comptes Rendus. Mathématique, Volume 338 (2004) no. 1, pp. 85-90.

Dans ce travail nous introduisons une classe de schémas numériques equilibrés au second ordre pour la solution de systèmes hyperboliques de lois de conservation. Nous donnons une technique générale pour construire ce type de schémas. Nous prouvons que ces schémas équilibrent au second ordre une grande classe de solutions stationaires, dans tout le domaine, excepté un sous-ensemble de mesure qui tend vers zéro lorsque la taille de la maille tend vers zéro. Nous présentons finalement une application aux équations de Saint-Venant qui montre les bonnes performances de quelques-uns des schémas présentés.

In this work we introduce a class of balanced numerical schemes, up to second order, for the solution of general non-homogeneous hyperbolic systems of conservation laws. We give a general technique to build such schemes. We also prove that they balance up to second order a large class of steady solutions in the whole domain but some subset whose measure tends to zero as the grid size decreases to zero. We finally present an application to Shallow Water equations that exhibit the good performances of some of the schemes introduced.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2003.11.008
Tomás Chacón Rebollo 1 ; Antonio Domı́nguez Delgado 2 ; Enrique D. Fernández Nieto 2

1 Dpto Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, c/Tarfia s/n, 41012 Sevilla, Spain
2 Dpto Matemática Aplicada I, Universidad de Sevilla, Avda. Reina Mercedes N. 2, 41012 Sevilla, Spain 
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     title = {Asymptotically balanced schemes for non-homogeneous hyperbolic systems {\textendash} application to the {Shallow} {Water} equations},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {85--90},
     publisher = {Elsevier},
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     year = {2004},
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Tomás Chacón Rebollo; Antonio Domı́nguez Delgado; Enrique D. Fernández Nieto. Asymptotically balanced schemes for non-homogeneous hyperbolic systems – application to the Shallow Water equations. Comptes Rendus. Mathématique, Volume 338 (2004) no. 1, pp. 85-90. doi : 10.1016/j.crma.2003.11.008. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2003.11.008/

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[3] T. Chacón; E.D. Fernández-Nieto; M. Gómez A flux-splitting solver for shallow water equations with source terms, Int. J. Numer. Methods Fluids, Volume 43 (2003), pp. 23-55

[4] E.D. Fernández-Nieto, Aproximación numérica de leyes de conservación hiperbólicas no homogéneas. Aplicación a las ecuaciones de Aguas Someras, Ph.D. Thesis Universidad de Sevilla, 2003

[5] 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

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