Hyperbolic problems with discontinuous coefficients and penalization of hyperbolic problems.
Problemes hyperboliques a coefficients discontinus et penalisation de problemes hyperboliques.
Résumé
This PhD Thesis is splitted into two parts.
1/ We are interested in the study of hyperbolic Cauchy problems with discontinuous coefficients.
We deal with discontinuities localized on one noncharacteristic hypersurface, also called interface, by using a vanishing viscosity approach. in different frameworks, we prove that the vanishing viscosity approach successfully singles out a unique solution. Different qualitative behaviors are shown to appear depending on the properties of the interface. in the case of systems, it is in general quite difficult to understand the properties of this interface.
2/ The goal here is to propose methods aimed at approximating the solutions of initial boundary value problems. More specifically, we propose domain penalization approaches for problems either well-posed in the sense of Friedrichs or well-posed in the sense of Kreiss.
The quality of the proposed methods are analyzed in terms of the boundary layers they generate.
1/ We are interested in the study of hyperbolic Cauchy problems with discontinuous coefficients.
We deal with discontinuities localized on one noncharacteristic hypersurface, also called interface, by using a vanishing viscosity approach. in different frameworks, we prove that the vanishing viscosity approach successfully singles out a unique solution. Different qualitative behaviors are shown to appear depending on the properties of the interface. in the case of systems, it is in general quite difficult to understand the properties of this interface.
2/ The goal here is to propose methods aimed at approximating the solutions of initial boundary value problems. More specifically, we propose domain penalization approaches for problems either well-posed in the sense of Friedrichs or well-posed in the sense of Kreiss.
The quality of the proposed methods are analyzed in terms of the boundary layers they generate.
Cette these se divise en deux parties.
1/ L'etude de problemes de Cauchy hyperboliques a coefficients discontinus.
Nous traitons de discontinuites localises sur une hypersurface non-caracteristique, representant une interface, au moyen d'une approche a viscosite evanescente. Dans differents cadres, nous montrons que l'approche a petite viscosite considere permet de selectionner une solution unique. Des comportements qualitatifs differents sont exhibes suivant la nature de l'interface. Dans le cadre de systemes, cerner la nature de l'interface s'avere etre en general delicat.
2/ Des methodes d' approximation de solutions de problemes aux limites hyperboliques bien poses au sens de Friedrichs ou de Kreiss sont donnees. Il s'agit de methodes de penalisation de domaines.
La qualite des methodes proposees est analysee en terme des couches limites engendrees.
1/ L'etude de problemes de Cauchy hyperboliques a coefficients discontinus.
Nous traitons de discontinuites localises sur une hypersurface non-caracteristique, representant une interface, au moyen d'une approche a viscosite evanescente. Dans differents cadres, nous montrons que l'approche a petite viscosite considere permet de selectionner une solution unique. Des comportements qualitatifs differents sont exhibes suivant la nature de l'interface. Dans le cadre de systemes, cerner la nature de l'interface s'avere etre en general delicat.
2/ Des methodes d' approximation de solutions de problemes aux limites hyperboliques bien poses au sens de Friedrichs ou de Kreiss sont donnees. Il s'agit de methodes de penalisation de domaines.
La qualite des methodes proposees est analysee en terme des couches limites engendrees.
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