Comptes Rendus
A formalism for the differentiation of conservation laws
[Un formalisme pour la dérivation des lois de conservations]
Comptes Rendus. Mathématique, Volume 335 (2002) no. 10, pp. 839-845.

On présente une méthode synthétique pour calculer les équations vérifiées par la dérivée par rapport à un paramètre de la solution v d'un système sous forme ∇·v=0. On montre, pour les équations de Burgers, Euler et Saint-Venant que la dérivée au sens usuel, mais interpretée au sens des distributions, contient les conditions de saut, c'est à dire les dérivées des conditions de transmission aux chocs. On retrouve ainsi les résultats de Godlewski–Raviart et al. que l'on étend aux équations d'Euler.

In this paper we present a synthetic method to differentiate with respect to a parameter partial differential equations in divergence form with shocks. We show that the usual derivatives contain the differentiated interface conditions if interpreted by the theory of distributions. We apply the method to three problems: the Burgers equation, the shallow water equations and Euler equations for fluids.

Accepté le :
Publié le :
DOI : 10.1016/S1631-073X(02)02574-8
Claude Bardos 1 ; Olivier Pironneau 1

1 Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, 175, rue du Chevaleret, Paris 75013, France
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Claude Bardos; Olivier Pironneau. A formalism for the differentiation of conservation laws. Comptes Rendus. Mathématique, Volume 335 (2002) no. 10, pp. 839-845. doi : 10.1016/S1631-073X(02)02574-8. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02574-8/

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[2] N. Di Cesare; O. Pironneau Shock sensitivity analysis, Comput. Fluid Dynamics J, Volume 9 (2000) no. 2

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[7] E. Godlewski; M. Olazabal; P.A. Raviart On the linearization of hyperbolic systems of conservation laws. Application to stability, Équations aux dérivées partielles et applications, Gauthier-Villars, Elsevier, Paris, 1998, pp. 549-570

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[10] C. Homescu, I. Navon, Numerical and theoretical considerations for sensitivity calculation of discontinuous. Systems Control Lett., to appear

[11] S. Jaouen, Étude mathématiques et numérique de stabilité pour des modèles hydrodynamiques, Thèse, Université Paris VI, 2001

[12] P.-L. Lions Mathematical Topics in Fluid Mechanics, Vol. 1, Oxford University Press, 1996

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