Study of a fully implicit 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 does not depend on the Debye length $\lambda$. This proves that the scheme is asymptotic preserving in the quasi-neutral limit $\lambda \to 0$.