First-order energy dissipative schemes in time are available in literature for the Poisson-Nernst-Planck (PNP) equations, but second-order ones are still in lack. This work proposes novel second-order discretization in time and finite volume discretization in space for modified PNP equations that incorporate effects arising from ionic steric interactions and dielectric inhomogeneity. A multislope method on unstructured meshes is proposed to reconstruct positive, accurate approximations of mobilities on faces of control volumes. Numerical analysis proves that the proposed numerical schemes are able to unconditionally ensure the existence of positive numerical solutions, original energy dissipation, mass conservation, and preservation of steady states at discrete level. Extensive numerical simulations are conducted to demonstrate numerical accuracy and performance in preserving properties of physical significance. Applications in ion permeation through a 3D nanopore show that the modified PNP model, equipped with the proposed schemes, has promising applications in the investigation of ion selectivity and rectification. The proposed second-order discretization can be extended to design temporal second-order schemes with original energy dissipation for a type of gradient flow problems with entropy.

翻译：关于泊松-能斯特-普朗克方程的时间一阶能量耗散格式已有文献报道，但二阶格式仍付阙如。本文针对考虑离子空间位阻效应与介电不均匀性的修正泊松-能斯特-普朗克方程，提出新颖的时间二阶离散格式及空间有限体积离散方法。通过提出非结构化网格上的多斜率重构方法，在控制体界面实现迁移率的正性高精度重构。数值分析证明，所提格式能在离散层面无条件保证数值解的正性存在、原始能量耗散、质量守恒及稳态保持。通过大量数值模拟验证了该格式的数值精度及其在保持物理重要性性质方面的性能。三维纳米孔离子渗透应用表明，配备所提格式的修正泊松-能斯特-普朗克模型在离子选择性及整流效应研究中具有广阔的应用前景。本文提出的二阶离散方法可推广至一类含熵的梯度流问题，设计具有原始能量耗散性质的时间二阶格式。