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Analyse et approximation numérique de systèmes hyperboliques de lois de conservation avec termes sources. Application aux équations d'Euler et à un modèle simplifié d'écoulements diphasiques.

Abstract : This thesis is devoted to the study of nonlinear hyperbolic systems of conservation laws with source terms allowed to become stiff. Applications come for the largest part from physical problems; for example B¨urgers or Buckley-Leverett equations and Euler-type systems endowed with pressure laws for a perfect gas or a simplified two-phase mixture. The first part deals with theoretical aspects of this subject. We focus on the one-dimensional case and consider the scalar equation and a special case of a reactive two-phase flow system. The main issue one has to manage with when establishing the strong convergence of a sequence of approximations by means of the compensated compactness theory in such cases is to find invariant regions in the phases space. For the scalar problem, it is sufficient that the source term preserves an uniform bound on the L∞ norm of the sequence. But it is less obvious for a system since invariant zones are rather complicated and may be perturbed due to the form of some forcing terms. The second part is concerned with stability criterions for numerical schemes involving sophisticated implicit time quadrature formulas. We first present an analysis of a half-discretization (method of lines) and derive sufficient conditions for the differential system to be well-posed. This brings quite natural restrictions to ensure the stability of a one-step implicit scheme: the time-step must be small in highly compressive regimes and near repulsive points of the source term. A total-variation analysis is then carried out to derive second order schemes free from oscillations around shocks for the largest possible time-steps. Numerical experiments are shown on fully nonlinear scalar problems. The third part is a direct sequel in order to fix some of the difficulties encountered before. The main objective is to build a new discretization which may handle an arbitrary stiff right hand-side without any influence (except the usual CFL condition) on the time-step. Following the ideas of J. Greenberg and A.-Y. LeRoux, we first derive a Godunov-type scalar scheme and get a convergence result with the help of BV estimates towards the entropy solution. Since the main feature is to handle the sources as non-conservative products, we decide to use the formalism proposed by G. DalMaso, P.G. LeFloch and F. Murat to extend these ideas to inhomogeneous systems. Following I. Toumi, we introduce Roe-type matrices along regularizing paths to obtain approximations showing nice properties. In fact, this is what is called well-balanced schemes, i.e. schemes which preserve theoretical first integrals of the motion. Numerical tests are based on the Euler system with geometrical source terms. The fourth (and last) part concerns the introduction of this non-conservative formulation in a flux-splitting type scheme. The aim is to build a very robust, well-balanced and easy-to-implement scheme. The generalized jump relations coming from the sources are detailed and the scheme is tested in a very wide range of problems: nozzle flows, two-phase with chemistry and damping, relaxation systems ... A chapter is also devoted to resonant cases: following Majda, we investigate the stationnary problem to validate our results. This work culminates with two-dimensional two-phase flow approximations which envolve rather complicated non-conservative relations treated by a convergent iterative process.
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Laurent Gosse. Analyse et approximation numérique de systèmes hyperboliques de lois de conservation avec termes sources. Application aux équations d'Euler et à un modèle simplifié d'écoulements diphasiques.. Analyse numérique [math.NA]. Université Paris Dauphine - Paris IX, 1997. Français. ⟨NNT : 97 PA09 0040⟩. ⟨tel-00773175⟩

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