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Contribution à l'approximation numérique des systèmes hyperboliques

Abstract : This work is devoted to several aspects of the numerical approximation of hyperbolic systems of conservation laws. The first part is dedicated to the derivation of high-order schemes on 2D unstructured meshes. We develop a new technique to reconstruct gradients based on two MUSCL schemes written on two overlapping meshes. This process increases the number of numerical unknowns, but it allows to approximate the solution very accurately. In the second part, we study the stability of high-order schemes. First, we show that the usual discrete entropy inequalities satisfied by high-order schemes are not relevant to ensure the good behaviour in the convergence regime. Therefore, we propose to extend the {\it a posteriori} limitation techniques to enforce the scheme to satisfy the required discrete entropy inequalities. In the last part, we focus on the derivation of well-balanced schemes for the Shallow water equations, the Ripa model and the Euler equations with gravity. We present several strategies leading to numerical schemes able to preserve all the steady states at rest. We also develop high-order extensions.
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https://tel.archives-ouvertes.fr/tel-00915482
Contributor : Vivien Desveaux Connect in order to contact the contributor
Submitted on : Sunday, December 8, 2013 - 11:58:39 AM
Last modification on : Sunday, November 27, 2022 - 2:48:05 PM
Long-term archiving on: : Saturday, March 8, 2014 - 10:16:15 PM

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  • HAL Id : tel-00915482, version 1

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Vivien Desveaux. Contribution à l'approximation numérique des systèmes hyperboliques. Analyse numérique [math.NA]. Université de Nantes, 2013. Français. ⟨NNT : ⟩. ⟨tel-00915482⟩

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