Convergence and long-time behavior of finite volumes for a generalized Poisson-Nernst-Planck system with cross-diffusion and size exclusion

We present a finite volume scheme for modeling the diffusion of charged particles, specifically ions, in constrained geometries using a degenerate Poisson-Nernst-Planck system with size exclusion yielding cross-diffusion. Our method utilizes a two-point flux approximation and is part of the exponentially fitted scheme framework. The scheme is shown to be thermodynamically consistent, as it ensures the decay of some discrete version of the free energy. Classical numerical analysis results – existence of discrete solution, convergence of the scheme as the grid size and the time step go to $0$ – follow. We also investigate the long-time behavior of the scheme, both from a theoretical and numerical point of view. Numerical simulations confirm our findings, but also point out some possibly very slow convergence towards equilibrium of the system under consideration.

💡 Research Summary

This paper addresses the numerical simulation of ion transport in confined geometries when both size‑exclusion (finite volume) effects and cross‑diffusion couplings are present. The authors start from a generalized Poisson‑Nernst‑Planck (GPNP) system in which the volume fractions of I ionic species, (u_i(x,t)), are constrained by a free‑space fraction (u_0=1-\sum_{i=1}^I u_i\ge0). The fluxes are written in the classical form
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