[Un formalisme pour la dérivation des lois de conservations]
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.
Publié le :
@article{CRMATH_2002__335_10_839_0,
author = {Bardos, Claude and Pironneau, Olivier},
title = {A formalism for the differentiation of conservation laws},
journal = {Comptes Rendus. Math\'ematique},
pages = {839--845},
publisher = {Elsevier},
volume = {335},
number = {10},
year = {2002},
doi = {10.1016/S1631-073X(02)02574-8},
language = {en},
url = {http://www.numdam.org/articles/10.1016/S1631-073X(02)02574-8/}
}
TY - JOUR AU - Bardos, Claude AU - Pironneau, Olivier TI - A formalism for the differentiation of conservation laws JO - Comptes Rendus. Mathématique PY - 2002 SP - 839 EP - 845 VL - 335 IS - 10 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/S1631-073X(02)02574-8/ DO - 10.1016/S1631-073X(02)02574-8 LA - en ID - CRMATH_2002__335_10_839_0 ER -
%0 Journal Article %A Bardos, Claude %A Pironneau, Olivier %T A formalism for the differentiation of conservation laws %J Comptes Rendus. Mathématique %D 2002 %P 839-845 %V 335 %N 10 %I Elsevier %U http://www.numdam.org/articles/10.1016/S1631-073X(02)02574-8/ %R 10.1016/S1631-073X(02)02574-8 %G en %F CRMATH_2002__335_10_839_0
Bardos, Claude; Pironneau, Olivier. A formalism for the differentiation of conservation laws. Comptes Rendus. Mathématique, Tome 335 (2002) no. 10, pp. 839-845. doi : 10.1016/S1631-073X(02)02574-8. http://www.numdam.org/articles/10.1016/S1631-073X(02)02574-8/
[1] Sensitivities to discontinuities in Darcy's Law, C. R. Acad. Sci. Paris, Série I, Volume 335 (2002), pp. 1-6
[2] Shock sensitivity analysis, Comput. Fluid Dynamics J, Volume 9 (2000) no. 2
[3] Adaptive finite element methods for parabolic problems. VI. Analytic semigroups, SIAM J. Numer. Anal, Volume 35 (1998), pp. 1315-1325
[4] Analytic adjoint solutions for the quasi-one-dimensional euler equations, J. Fluid Mech, Volume 426 (2001), pp. 327-345
[5] E. Godlewski, P.-A. Raviart, An introduction to the linearized stability of solutions of nonlinear hyperbolic systems of conservation laws, UPMC-J.-L. Lions Laboratory report R0003, 2000
[6] E. Godlewski, P.-A. Raviart, The linearized stability of solutions of nonlinear hyperbolic systems of conservation laws: A general numerical approach. of conservation laws, UPMC-J.-L. Lions Laboratory report R9850, 1998
[7] 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
[8] Numerical Methods for Nonlinear Variational Problems, Springer-Verlag, New York, 1984
[9] Evaluating Derivatives, Principles and Techniques of Algorithmic Differentiation, Frontiers Appl. Math, 19, SIAM, 2000
[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] Mathematical Topics in Fluid Mechanics, Vol. 1, Oxford University Press, 1996
[13] The Stability of Multi-dimensional Shock Fronts, Mem. Amer. Math. Soc, 281, American Mathematical Society, Providence, RI, 1983
[14] Écoulements de fluide : compacité par entropie, Masson, Paris, 1989
[15] Hyperbolic and viscous conservation laws, CBMS-NSF Regional Conference Series in Applied Mathematics, 72, SIAM, Philadelphia, PA, 2000
Cité par Sources :
