An asymptotic preserving scheme with the maximum principle for the $ {M}_{1}$ model on distorded meshes

Christophe Buet; Bruno Després; Emmanuel Franck

doi:10.1016/j.crma.2012.07.002

Numerical Analysis

An asymptotic preserving scheme with the maximum principle for the

M_{1}

model on distorded meshes
[Un nouveau schéma préservant lʼasymptotique avec le principe du maximum pour le modèle

M_{1}

sur maillages quelconques]

Présenté par : Olivier Pironneau

Christophe Buet ¹ ; Bruno Després ² ; Emmanuel Franck ^2,¹

¹ CEA, DAM, DIF, 91297 Arpajon, France
² UPMC Univ. 06, UMR 7598, laboratoire J.L. Lions, 75005 Paris, France

Comptes Rendus. Mathématique, Volume 350 (2012) no. 11-12, pp. 633-638.

Résumé
Abstract

Dans cette Note, nous montrons quʼun nouveau schéma introduit dans Buet et al. (2011) [5] pour le modèle à deux moments non linéaire $M_{1}$ de lʼéquation de transport et qui est compatible avec la limite de diffusion (schéma AP) sur maillage quelconque vérifie aussi le principe du maximum. Lʼidée consiste à réécrire le modèle comme un système de la dynamique des gaz, puis à utiliser un schéma Eulerien nodal, dérivé dʼun schéma Lagrange + projection couplé à une extension multidimensionnelle, developpée dans Buet et al. (2012) [6], de la méthode de Jin et Levermore (1996) [9] pour lʼéquation de la chaleur hyperbolique. Après la présentation du schéma on donne les preuves dʼentropie et de principe du maximum. Pour finir on présente des résultats numériques pour des maillages déformés triangulaires et quadrangulaires qui montrent notamment lʼordre deux dans le régime de diffusion.

In this Note, we show that a recent scheme introduced by Buet et al. (2011) [5] for the nonlinear two moments $M_{1}$ model of linear transport and which captures correctly the diffusion limit on distorded meshes (AP scheme) also possesses the maximum principle. The main idea of the design of this scheme is to rewrite the model as a gas dynamics model and to use an Eulerian scheme, derived from a Lagrange + remap scheme. To obtain the AP property we use the multidimensional extension, developed by Buet et al. (2012) [6], of the Jin and Levermore (1996) procedure [9] for the hyperbolic heat equation. We will show that this scheme is entropic which ensures the maximum principle of the $M_{1}$ model. More we present some numerical results, on distorted quadrangular and triangular meshes which show that the scheme is second order in the diffusive regime.

Reçu le : 2012-04-10
Accepté le : 2012-07-04
Publié le : 2012-06-01

DOI : 10.1016/j.crma.2012.07.002

Affiliations des auteurs :

Christophe Buet ¹ ; Bruno Després ² ; Emmanuel Franck ^{2, 1}

¹ CEA, DAM, DIF, 91297 Arpajon, France
² UPMC Univ. 06, UMR 7598, laboratoire J.L. Lions, 75005 Paris, France

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     author = {Christophe Buet and Bruno Despr\'es and Emmanuel Franck},
     title = {An asymptotic preserving scheme with the maximum principle for the $ {M}_{1}$ model on distorded meshes},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {633--638},
     publisher = {Elsevier},
     volume = {350},
     number = {11-12},
     year = {2012},
     doi = {10.1016/j.crma.2012.07.002},
     language = {en},
}

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Christophe Buet; Bruno Després; Emmanuel Franck. An asymptotic preserving scheme with the maximum principle for the $ {M}_{1}$ model on distorded meshes. Comptes Rendus. Mathématique, Volume 350 (2012) no. 11-12, pp. 633-638. doi : 10.1016/j.crma.2012.07.002. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.07.002/

Bibliographie
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Xavier Blanc; Philippe Hoch; Clément Lasuen An asymptotic preserving scheme for the $M_{1}$ model on polygonal and conical meshes, Calcolo, Volume 61 (2024) no. 2 | DOI:10.1007/s10092-024-00574-4
Q. Schmid; Y. Mesri; E. Hachem; R. Codina Stabilized mixed finite element method for the M1 radiation model, Computer Methods in Applied Mechanics and Engineering, Volume 335 (2018), p. 69 | DOI:10.1016/j.cma.2018.01.046
F. Blachère; R. Turpault An admissibility and asymptotic preserving scheme for systems of conservation laws with source term on 2D unstructured meshes with high-order MOOD reconstruction, Computer Methods in Applied Mechanics and Engineering, Volume 317 (2017), p. 836 | DOI:10.1016/j.cma.2017.01.012
Emmanuel Franck; Laura S. Mendoza Finite Volume Scheme with Local High Order Discretization of the Hydrostatic Equilibrium for the Euler Equations with External Forces, Journal of Scientific Computing, Volume 69 (2016) no. 1, p. 314 | DOI:10.1007/s10915-016-0199-4
Christophe Buet; Bruno Després; Emmanuel Franck Asymptotic Preserving Schemes on Distorted Meshes for Friedrichs Systems with Stiff Relaxation: Application to Angular Models in Linear Transport, Journal of Scientific Computing, Volume 62 (2015) no. 2, p. 371 | DOI:10.1007/s10915-014-9859-4
Emmanuel Franck Modified Finite Volume Nodal Scheme for Euler Equations with Gravity and Friction, Finite Volumes for Complex Applications VII-Methods and Theoretical Aspects, Volume 77 (2014), p. 285 | DOI:10.1007/978-3-319-05684-5_27
Laurent Gosse The Special Case of 2-Velocity Kinetic Models, Computing Qualitatively Correct Approximations of Balance Laws, Volume 2 (2013), p. 137 | DOI:10.1007/978-88-470-2892-0_8

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