Study of a Finite Volume Scheme for the Drift-Diffusion System. Asymptotic Behavior in the Quasi-Neutral Limit

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 do not depend on the Debye length λ. This proves that the scheme is asymptotic preserving in the quasi-neutral limit λ → 0.