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Schémas numériques d'ordre élevé et préservant l'asymptotique pour l'hydrodynamique radiative

Abstract : The aim of this work is to design a high-order and explicit finite volume scheme for specific systems of conservation laws with source terms. Those systems may degenerate into diffusion equations under some compatibility conditions. The degeneracy is observed with large source term and/or with late-time. For instance, this behaviour can be seen with the isentropic Euler model with friction or with the M1 model for radiative transfer, or with the radiation hydrodynamics model. We propose a general theory to design a first-order asymptotic preserving scheme (in the sense of Jin) to follow this degeneracy. The scheme is proved to be stable and consistent under a classical hyperbolic CFL condition in both hyperbolic and diffusive regimes, for any 2D unstructured mesh. Moreover, we justify that the developed scheme also preserves the set of admissible states in all regimes, which is mandatory to conserve physical solutions. This construction is achieved by using the non-linear scheme of Droniou and Le Potier as a target scheme for the diffusive equation, which gives the form of the global scheme for the complete system of conservation laws. Then, the high-order scheme is constructed with polynomial reconstructions and the MOOD paradigm as a limiter. The main difficulties are the preservation of the set of admissible states in both regimes on unstructured meshes and to deal with the high-order polynomial reconstruction in the diffusive limit without losing the asymptotic preserving property. Numerical results are provided to validate the scheme in all regimes, with the first and high-order versions.
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https://tel.archives-ouvertes.fr/tel-01373877
Contributor : Florian Blachère <>
Submitted on : Thursday, September 29, 2016 - 12:22:44 PM
Last modification on : Tuesday, December 8, 2020 - 10:52:38 AM
Long-term archiving on: : Friday, December 30, 2016 - 1:10:19 PM

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Florian Blachère. Schémas numériques d'ordre élevé et préservant l'asymptotique pour l'hydrodynamique radiative. Analyse numérique [math.NA]. Université Nantes; Université Bretagne Loire, 2016. Français. ⟨tel-01373877⟩

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