no. 7
Numerical Analysis/Mathematical Physics
Numerical approximation of a quantum drift-diffusion model
[Approximation numérique d'un modèle de dérive-diffusion quantique.]

Présenté par : Philippe G. Ciarlet

Gallego, Samy1 ; Méhats, Florian1
1 Mathématiques pour l'industrie et la physique (UMR 5640), université Paul Sabatier, 118, route de Narbonne, 31062 Toulouse cedex 4, France
Comptes Rendus. Mathématique, Tome 339 (2004) no. 7, pp. 519-524.

Cette Note est consacrée à la discrétisation et à la simulation numérique d'un modèle de dérive-diffusion quantique qui a été dérivé récemment. Nous définissons un schéma numérique implicite dont la résolution se ramène à un problème de minimisation convexe. Par ailleurs, ce schéma préserve les propriétés physiques vérifiées par le modèle continu : conservation de la charge totale, positivité de la densité et dissipation d'une entropie. Enfin, nous illustrons ces propriétés à l'aide de simulations numériques.

This Note is devoted to the discretization and numerical simulation of a new quantum drift-diffusion model that was recently derived. We define an implicit numerical scheme which is equivalent to a convex minimization problem and which preserves the physical properties of the continuous model: charge conservation, positivity of the density and dissipation of an entropy. We illustrate these results by some numerical simulations.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2004.07.022
Gallego, Samy 1 ; Méhats, Florian 1

1 Mathématiques pour l'industrie et la physique (UMR 5640), université Paul Sabatier, 118, route de Narbonne, 31062 Toulouse cedex 4, France 
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Gallego, Samy; Méhats, Florian. Numerical approximation of a quantum drift-diffusion model. Comptes Rendus. Mathématique, Tome 339 (2004) no. 7, pp. 519-524. doi : 10.1016/j.crma.2004.07.022. http://www.numdam.org/articles/10.1016/j.crma.2004.07.022/

[1] Ancona, M.G.; Iafrate, G.J. Quantum correction of the equation of state of an electron gas in a semiconductor, Phys. Rev. B, Volume 39 (1989), pp. 9536-9540

[2] Ben Abdallah, N.; Unterreiter, A. On the stationary quantum drift-diffusion model, Z. Angew. Math. Phys., Volume 49 (1998) no. 2, pp. 251-275

[3] P. Degond, F. Méhats, C. Ringhofer, Quantum energy-transport and drift-diffusion models, submitted for publication

[4] P. Degond, F. Méhats, C. Ringhofer, Quantum hydrodynamic models derived from entropy principle, Contemp. Math., in press

[5] Degond, P.; Ringhofer, C. Quantum moment hydrodynamics and the entropy principle, J. Statist. Phys., Volume 112 (2003) no. 3/4, pp. 587-628

[6] Degond, P.; Ringhofer, C. Binary quantum collision operators conserving mass momentum and energy, C. R. Acad. Sci. Paris, Ser. I, Volume 336 (2003) no. 9, pp. 785-790

[7] S. Gallego, F. Méhats, Entropic discretization of a quantum drift-diffusion model, submitted for publication

[8] Jüngel, A. Quasi-Hydrodynamic Semiconductor Equations, Progress in Nonlinear Differential Equations, Birkhäuser, 2001

[9] Kaiser, H.-C.; Rehberg, J. About a stationary Schrödinger–Poisson system with Kohn–Sham potential in a bounded two- or three-dimensional domain, Nonlinear Anal., Volume 41 (2000), pp. 33-72

[10] Levermore, C.D. Moment closure hierarchies for kinetic theories, J. Statist. Phys., Volume 83 (1996) no. 5/6, pp. 1021-1065

[11] Nier, F. A stationary Schrödinger–Poisson system arising from the modelling of electronic devices, Forum Math., Volume 2 (1990) no. 5, pp. 489-510

[12] Nier, F. A variational formulation of Schrödinger–Poisson systems in dimension d⩽3, Commun. Partial Differential Equations, Volume 18 (1993) no. 7/8, pp. 1125-1147

[13] Pinnau, R. The linearized transient quantum drift diffusion model – stability of stationary states, Z. Angew. Math. Mech. (ZAMM), Volume 80 (2000) no. 5, pp. 327-344

[14] Reed, M.; Simon, B.; Reed, M.; Simon, B. Methods of Modern Mathematical Physics, vol. II, Fourier Analysis, Self-Adjointness, Methods of Modern Mathematical Physics, vol. IV, Analysis of Operators, Academic Press, New York, 1975

[15] Serre, D. Matrices. Theory and Applications, Grad. Texts in Math., vol. 216, Springer-Verlag, New York, 2002 (translated from the 2001 French original)

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