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.

翻译：本文提出了一种有限体积格式，用于模拟带电粒子（特别是离子）在受限几何结构中的扩散过程，该模型采用具有尺寸排阻效应并产生交叉扩散的退化型泊松-能斯特-普朗克系统。本方法基于两点通量近似，属于指数拟合格式框架的组成部分。该格式被证明具有热力学一致性，因为它保证了离散版本自由能的衰减特性。经典的数值分析结论——包括离散解的存在性、以及当网格尺寸与时间步长趋于 $0$ 时格式的收敛性——均得以验证。我们还从理论与数值计算两个角度研究了该格式的长时间行为。数值模拟结果不仅验证了我们的理论发现，同时揭示了所研究系统在趋于平衡过程中可能存在的极端缓慢收敛现象。