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

TL;DR

This paper compares standard and asymptotic-preserving time discretizations for the Poisson-Nernst-Planck system, demonstrating that IMEX schemes offer stable solutions across various Debye lengths, especially in the quasi-neutral limit.

Contribution

The study validates and compares different time discretization methods, highlighting the advantages of IMEX schemes in handling the quasi-neutral limit of the PNP system.

Findings

IMEX schemes are asymptotically stable for all Debye lengths.

Standard methods face severe stability issues at small Debye lengths.

The paper demonstrates the effectiveness of asymptotic-preserving schemes 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.