Développement et analyse de méthodes adaptatives pour les équations de transport

Abstract : This thesis focuses on adaptive approximation of two nonlinear transport problems, namely the Vlasov-Poisson system and the scalar conservation laws. In a semi-lagrangian approach, we propose a new adaptive scheme for the first one, in which the mesh is, at each time step, first predicted in a very simple way, then corrected by a classical algorithm. In order to extend the W2,1(R2) semi-norm to piecewise affine functions, the notion of total curvature is introduced and employed in a rigorous analysis to obtain a priori error estimates that establish the convergence of this scheme in the L∞ metric, while a partial result of complexity is proposed. As scalar conservation laws may not be approximated in the same metric, we consider the uniform Hausdorff distance which appears as a natural substitute for the L∞ one, and show that the solutions are stable with respect to this distance. Equipped with this new result, we prove a high order adaptive approximation theorem for these equations.
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https://tel.archives-ouvertes.fr/tel-00129013
Contributor : Martin Campos Pinto <>
Submitted on : Monday, February 5, 2007 - 7:02:56 PM
Last modification on : Wednesday, May 15, 2019 - 3:38:16 AM
Long-term archiving on : Tuesday, April 6, 2010 - 11:11:08 PM

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  • HAL Id : tel-00129013, version 1

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Martin Campos Pinto. Développement et analyse de méthodes adaptatives pour les équations de transport. Mathématiques [math]. Université Pierre et Marie Curie - Paris VI, 2005. Français. ⟨tel-00129013⟩

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