Standard versus Asymptotic Preserving Time Discretizations for the Poisson-Nernst-Planck System in the Quasi-Neutral Limit

Abstract

In this paper, we investigate the correlated diffusion of two ion species governed by a Poisson-Nernst-Planck (PNP) system. Here we further validate the numerical scheme recently proposed in \cite{astuto2025asymptotic}, where a time discretization method was shown to be Asymptotic-Preserving (AP) with respect to the Debye length. For vanishingly Debye lengths, the so called Quasi-Neutral limit can be adopted, reducing the system to a single diffusion equation with an effective diffusion coefficient \cite{CiCP-31-707}. Choosing small, but not negligible, Debye lengths, standard numerical methods suffer from severe stability restrictions and difficulties in handling initial conditions. IMEX schemes, on the other hand, are proved to be asymptotically stable for all Debye lengths, and do not require any assumption on the initial conditions. In this work, we compare different time discretizations to show their asymptotic behaviors.