An asymptotic preserving and accurate scheme for multiscale Poisson-Nernst-Planck (MPNP) system

In this paper, we propose and validate a two-species Multiscale model for a Poisson-Nernst-Planck (PNP) system, focusing on the correlated motion of positive and negative ions under the influence of a trap. Specifically, we aim to model surface traps whose attraction range, of length $\delta$, is much smaller then the scale of the problem. The physical setup we refer to is an anchored gas drop (bubble) surrounded by a diffusive flow of charged surfactants (ions). When the diffusingsurfactants reach the surface of the trap, the anions are adsorbed. As in our previous works \cite{astuto2023multiscale,ASTUTO2023111880,astuto2025time,astuto2024high}, the effect of the attractive potential is replaced by a suitable boundary condition derived by mass conservation and asymptotic analysis. The novelty of this work is the extension of the model proposed in \cite{astuto2023multiscale}, now incorporating the influence of both carriers -- positive and negative ions -- simultaneously, which is often neglected in traditional approaches that treat ion species independently.In the second part of the paper, we address the treatment of the Coulomb interaction between carriers. When the Debye length $\lambda_D$ (proportional to a small parameter $\varepsilon$) is very small, one can adopt the so-called Quasi-Neutral limit, which significantly simplifies the system, reducing it to a diffusion equation for a single carriers with effective diffusion coefficient \cite{jungel,CiCP-31-707}. This approach, while simplifying the mathematical model, does not capture the effects of non negligible values of $\varepsilon$. When the Debye length is small but not negligible, it may be very expensive to capture the small deviation from the Quasi-Neutral limit by standard methods in the literature. One of the objectives of this work is to develop an \textit{Asymptotic Preserving} (AP) second order numerical scheme that works for all Debye lengths and becomes a consistent discretization of the Quasi Neutral limit as $\varepsilon \to 0$, with no stability restriction on the time step.%Our AP scheme is designed to handle both regimes: the full system when the Debye length is small but non-negligible, and the Quasi-Neutral regime as the Debye length approaches zero.Furthermore, the numerical scheme we propose is also \textit{Asymptotic Accurate} (AA), which means that it preserves second order accuracy in the Quasi-Neutral limit.