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tel-00656013, version 1 - 3 Jan 2012
tel-00656013, version 1 - 3 Jan 2012 Abstract This work is concerned with different aspects of multiscale partial differential equations. For three distinct problems, we address questions of modelling and discretization thanks to the observation of the multiplicity of scales, time or space. So we propose in the first part a model of approximation of a fluid with a free surface, say an ocean. The derivation of our multilayer shallow water type model is based on an analysis of the different space scales generally observed in geophysical flows, in particular the so-called shallow water assumption. We obtain an existence and uniqueness result of local in time solution. Next we propose a finite volume scheme and numerical simulations in order to validate our model. In the second part, we study a hyperbolic relaxation problem, initially motivated by the kinetic theory of gaz. Different time scales appear through the competition between a transport phenomenon and a relaxation one, to a local equilibrium. 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. Then we present some numerical simulations. The last part deals with a blood flow model in a stented artery. We consider a Stokes problem stated in a multiscale space domain, that is a macroscopic box (the artery) containing a microscopic roughness (the stent). In order to avoid expensive simulations when discretizing the whole rough domain, we perform a Chapman-Enskog type development of the solution and derive an implicit wall law on the boundary of the smooth domain. Error estimates are shown and numerical illustrations of the results are presented. Key words: multilayer shallow water model, hyperbolic systems, conservation laws, finite volumes, relaxation, source term, stiff, total variation diminishing, asymptotic preserving schemes, boundary layer, wall-laws.
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