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Modélisation, analyse mathématique et simulations numériques de quelques problèmes aux dérivées partielles multi-échelles

Abstract : This work is concerned with different aspects of multiscale partial differential equations. For three problems, we address questions of modelling and discretization thanks to the observation of the multiplicity of scales, time or space. We propose in the first part a model of approximation of a fluid with a free surface (ocean). The derivation of our multilayer shallow water type model is based on the analysis of the different space scales generally observed in geophysical flows, precisely the 'shallow water' assumption. We obtain an existence and uniqueness result of local in time solution and propose a finite volume scheme and numerical simulations. Next we study a hyperbolic relaxation problem, motivated by the kinetic theory of gaz. Adopting an Asymptotic Preserving strategy of discretization, we build and analyze a numerical scheme. The convergence is proved for any value of the relaxation parameter, as well as the consistency with the equilibrium problem, thanks to error estimates. We present some numerical simulations. The last part deals with a blood flow model in a stented artery. We consider a Stokes problem in a multiscale space domain, that is a macroscopic box (the artery) containing a microscopic roughness (the stent). To avoid expensive simulations when discretizing the whole rough domain, we perform a Chapman-Enskog type expansion of the solution and derive an implicit wall law on the boundary of the smooth domain. Error estimates are shown and numerical simulations are presented
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https://tel.archives-ouvertes.fr/tel-00656013
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Submitted on : Monday, February 17, 2014 - 10:28:23 AM
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  • HAL Id : tel-00656013, version 2

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Amélie Rambaud. Modélisation, analyse mathématique et simulations numériques de quelques problèmes aux dérivées partielles multi-échelles. Mathématiques générales [math.GM]. Université Claude Bernard - Lyon I, 2011. Français. ⟨NNT : 2011LYO10317⟩. ⟨tel-00656013v2⟩

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