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Ecole Polytechnique X (25/11/2002), LeFloch Philippe (Dir.)
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Bilans d'entropie discrets dans l'approximation numerique des chocs non classiques. Application aux equations de Navier-Stokes multi-pression 2D et a quelques systemes visco-capillaires
Christophe Chalons1

La presente recherche doctorale en analyse numerique (et calcul scientifique) aborde le probleme du controle de la dissipation d'entropie numerique associee a une discretisation donnee. Cette problematique constitue un veritable challenge numerique non encore completement resolu a ce jour.

Le travail se decompose en deux parties principales dont les caracteristiques sont reellement differentes.

La premiere partie concerne l'approximation numerique des solutions (instationnaires en 1D et stationnaires en 2D) du systeme des equations de Navier-Stokes a plusieurs pressions independantes. Ce systeme est hyperbolique et possede des champs vraiment non lineaires sous des hypotheses classiques, mais s'ecrit naturellement sous forme non conservative.

La deuxieme partie est dediee a l'approximation numerique des solutions instationnaires en 1D de quelques systemes de lois de conservation de type soit hyperbolique mais dont les champs possedent un defaut de vraiment non linearite, ou soit mixte hyperbolique-elliptique.

Dans toutes ces situations motivees par des applications physiques concretes, le controle de la dissipation d'entropie joue un role déterminant dans la caracterisation des solutions recherchees. Les schemas numeriques proposes dans ce manuscrit sont obtenus par une analyse fine des bilans d'entropie associes.
1:  CMAP - Centre de Mathématiques Appliquées
Equations de Navier-Stokes multi-pression – Methode de Relaxation – schemas volumes finis explicites et implicites – relations de saut generalisees – bilans d'entropie discrets – systemes non conservatifs – systemes visco-capillaires – systeme hyperbolique – systeme hyperbolique-elliptique – chocs non classiques – fluides de van der Waals – transitions de phase subsoniques

In this thesis (numerical analysis and scientific computation areas), we are interested in
the knowledge of the numerical entropy dissipation associated with a given numerical scheme. It turns out that generally speaking, this problem is still open while being of crucial interest in many applications.

Our study is composed of two distinct parts.

The first one is concerned with the numerical approximation of the (unsteady in 1D and steady in 2D) solutions of the Navier-Stokes equations involving several independent pressure laws. As in the usual setting of a single pressure law, this system is hyperbolic with genuinely non linear associated fields under classical assumptions.
However, it naturally writes in non conservation form.

The second one deals with the numerical approximation of the unsteady solutions (in 1D) of several systems of conservation laws which are either hyperbolic but with non genuinely non linear (and non linearly degenerate) fields, or mixed hyperbolic-elliptic.

Several models from the physics enter the present framework for which the entropy dissipation plays an important role in the selection of the physical solution. Here we propose several relevant numerical schemes designed on the basis of a precise study of the associated entropy dissipation.

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