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Comptes Rendus Mathématique
Volume 354, n° 1
pages 39-43 (janvier 2016)
Doi : 10.1016/j.crma.2015.09.020
Received : 30 Mars 2015 ;  accepted : 24 September 2015
A generalization of the quantum Bohm identity: Hyperbolic CFL condition for Euler–Korteweg equations
Généralisation de l'identité de Bohm quantique : condition CFL hyperbolique pour équations d'Euler–Korteweg
 

Didier Bresch a, 1 , Frédéric Couderc b , Pascal Noble b, 2 , Jean-Paul Vila b
a LAMA – UMR5127 CNRS, bâtiment Le Chablais, campus scientifique, 73376 Le Bourget-du-Lac, France 
b IMT, INSA Toulouse, 135, avenue de Rangueil, 31077 Toulouse cedex 9, France 

Abstract

In this note, we propose a surprising and important generalization of the quantum Bohm potential identity. This formula allows us to design an original conservative extended formulation of Euler–Korteweg systems and the construction of a numerical scheme with entropy stability property under a hyperbolic CFL condition in the multi-dimensional setting. To the authors' knowledge, this generalization of the quantum Bohm identity strongly improves what is already known for simulation of such a dispersive system and is also important for theoretical studies on compressible Navier–Stokes equations with degenerate viscosities.

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Résumé

Dans cette note, on propose une importante généralisation de l'identité dite du potentiel de Bohm quantique. Cette dernière permet de définir une formulation augmentée des systèmes d'Euler–Korteweg, qui est sous forme conservative dans le cas multi-dimensionnel. Une conséquence très importante de cette formulation est la construction de schémas avec stabilité entropique sous condition CFL hyperbolique du système d'Euler–Korteweg. Cette généralisation de l'identité de Bohm évite donc le développement d'ondes parasites pour ces systèmes de type dispersif et est aussi importante, par exemple, dans l'étude des équations de Navier–Stokes compressibles à viscosités dégénérées.

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1  Research of D.B. was partially supported by the ANR project DYFICOLTI ANR-13-BS01-0003-01.
2  Research of P.N. was partially supported by the ANR project BoND ANR-13-BS01-0009-01.


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