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Méthodes de Galerkin stochastiques adaptatives pour la propagation d'incertitudes paramétriques dans les modèles hyperboliques

Abstract : This work is concerned with stochastic Galerkin methods for hyperbolic systems involving uncertain data with known distribution functions parametrized by random variables. We are interested in problems where a shock appears almost surely in finite time. In this case, the solution exhibits discontinuities in the spatial and in the stochastic domains. A Finite Volume scheme is used for the spatial discretization and a Galerkin projection based on piecewise poynomial approximation is used for the stochastic discretization. A Roe-type solver with an entropy correction is proposed for the Galerkin system, using an original technique to approximate the absolute value of the Roe matrix and an adaptation of the Dubois and Mehlman entropy corrector. Although this method deals with complex situations, it remains costly because a very fine stochastic discretization is needed to represent the solution in the vicinity of discontinuities. This fact calls for adaptive strategies. As discontinuities are localized in space and time, stochastic representations depending on space and time are proposed. This methodology is formulated in a multiresolution context based on the concept of binary trees for the stochastic discretization. The adaptive enrichment and coarsening steps are based on multiresolution analysis criteria. In the multidimensional case, an anisotropy of the adaptive procedure is proposed. The method is tested on the Euler equations in a shock tube and on the Burgers equation in one and two stochastic dimensions
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Submitted on : Wednesday, February 27, 2013 - 5:47:16 PM
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Julie Tryoen. Méthodes de Galerkin stochastiques adaptatives pour la propagation d'incertitudes paramétriques dans les modèles hyperboliques. Mathématiques générales [math.GM]. Université Paris-Est, 2011. Français. ⟨NNT : 2011PEST1054⟩. ⟨pastel-00795322⟩



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