Numerical Analysis
An asymptotic preserving scheme with the maximum principle for the M1 model on distorded meshes
Comptes Rendus. Mathématique, Volume 350 (2012) no. 11-12, pp. 633-638.

In this Note, we show that a recent scheme introduced by Buet et al. (2011) [5] for the nonlinear two moments M1 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 M1 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.

Dans cette Note, nous montrons quʼun nouveau schéma introduit dans Buet et al. (2011) [5] pour le modèle à deux moments non linéaire M1 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.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2012.07.002
Buet, Christophe 1; Després, Bruno 2; Franck, Emmanuel 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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Buet, Christophe; Després, Bruno; Franck, Emmanuel. 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. http://www.numdam.org/articles/10.1016/j.crma.2012.07.002/

[1] Berthon, C.; Buet, C.; Coulombel, J.-F.; Desprès, B.; Dubois, J.; Goudon, T.; Morel, J.E.; Turpault, R. Mathematical Models and Numerical Methods for Radiative Transfer, Panoramas et Synthèses, vol. 28, 2009

[2] Berthon, C.; Charrier, P.; Dubroca, B. An HLLC scheme to solve the M1 model of radiative transfer in two space dimensions, J. Sci. Comput., Volume 31 (2007) no. 3, pp. 347-389

[3] Berthon, C.; Dubois, J.; Dubroca, B.; Nguyen-Bui, T.-H.; Turpault, R. A free streaming contact preserving scheme for the M1 model, Adv. Appl. Math. Mech., Volume 3 (2010), pp. 259-285

[4] Buet, C.; Després, B. Grey radiative hydrodynamics; hierarchy of models and numerical approximation, Mathematical Models and Numerical Methods for Radiative Transfer, Panor. Synthèses, vol. 28, SMF, Paris, 2009

[5] C. Buet, B. Després, E. Franck, Asymptotic preserving finite volumes discretization for non-linear moment model on unstructured meshes, in: Proceedings FVCA6, 2011.

[6] Buet, C.; Després, B.; Franck, E. Design of asymptotic preserving finite volume schemes for the hyperbolic heat equation on unstructured meshes Numer. Math. (2012) | DOI

[7] Carré, G.; Del Pino, S.; Després, B.; Labourasse, E. A cell-centered Lagrangian hydrodynamics scheme on general unstructured meshes in arbitrary dimension, J. Comput. Phys., Volume 228 (2009) no. 14, pp. 5160-5518

[8] Gosse, L.; Toscani, G. An asymptotic-preserving well-balanced scheme for the hyperbolic heat equations, C. R. Acad. Sci. Paris, Ser. I, Volume 334 (2002), pp. 337-342

[9] Jin, S.; Levermore, C.D. Numerical schemes for hyperbolic conservation laws with stiff relaxation terms, J. Comput. Phys., Volume 126 (1996), pp. 449-467

[10] Levermore, C.D. Relating Eddington factors to flux limiters, J. Quant. Spectrosc. Radiat. Transfer, Volume 31 (1984) no. 2, pp. 149-160

[11] Maire, P.-H.; Abgrall, R.; Breil, J.; Ovadia, J. A cell-centered Lagrangian scheme for two-dimensional compressible flow problems, SIAM J. Sci. Comput., Volume 29 (2007) no. 4, pp. 1781-1824

[12] Nessyahu, H.; Tadmor, E. The convergence rate of approximate solutions for nonlinear scalar conservation laws, SIAM J. Numer. Anal., Volume 29 (1991), pp. 1505-1519

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