Study of a fully implicit scheme for the drift-diffusion system. Asymptotic behavior in the quasi-neutral limit

Marianne Bessemoulin-Chatard (1) , Claire Chainais-Hillairet (2, 3) , Marie-Hélène Vignal (4)

1 MMCS - Modélisation mathématique, calcul scientifique
2 LPP - Laboratoire Paul Painlevé - UMR 8524
3 MEPHYSTO-POST - Méthodes quantitatives pour les modèles aléatoires de la physique
4 IMT - Institut de Mathématiques de Toulouse UMR5219

In this paper, we are interested in the numerical approximation of the classical time-dependent drift-diffusion system near quasi-neutrality. We consider a fully implicit in time and finite volume in space scheme, where the convection-diffusion fluxes are approximated by Scharfetter-Gummel fluxes. We establish that all the a priori estimates needed to prove the convergence of the scheme does not depend on the Debye length $\lambda$. This proves that the scheme is asymptotic preserving in the quasi-neutral limit $\lambda \to 0$.

Mots clés: quasi-neutral limit; drift-diffusion; entropy method; asymptotic preserving schemes; finite volume

Domaines: Analyse numérique [math.NA]

Marianne Bessemoulin-Chatard, Claire Chainais-Hillairet, Marie-Hélène Vignal. Study of a fully implicit scheme for the drift-diffusion system. Asymptotic behavior in the quasi-neutral limit. SIAM Journal on Numerical Analysis, 2014, 52 (4), pp.1666-1691. ⟨10.1137/130913432⟩. ⟨hal-00801912v2⟩