In this paper we propose and analyze a finite difference numerical scheme for the Poisson-Nernst-Planck equation (PNP) system. To understand the energy structure of the PNP model, we make use of the Energetic Variational Approach (EnVarA), so that the PNP system could be reformulated as a non-constant mobility $H^{-1}$ gradient flow, with singular logarithmic energy potentials involved. To ensure the unique solvability and energy stability, the mobility function is explicitly treated, while both the logarithmic and the electric potential diffusion terms are treated implicitly, due to the convex nature of these two energy functional parts. The positivity-preserving property for both concentrations, $n$ and $p$, is established at a theoretical level. This is based on the subtle fact that the singular nature of the logarithmic term around the value of $0$ prevents the numerical solution reaching the singular value, so that the numerical scheme is always well-defined. In addition, an optimal rate convergence analysis is provided in this work, in which many highly non-standard estimates have to be involved, due to the nonlinear parabolic coefficients. The higher order asymptotic expansion (up to third order temporal accuracy and fourth order spatial accuracy), the rough error estimate (to establish the $\ell^\infty$ bound for $n$ and $p$), and the refined error estimate have to be carried out to accomplish such a convergence result. In our knowledge, this work will be the first to combine the following three theoretical properties for a numerical scheme for the PNP system: (i) unique solvability and positivity, (ii) energy stability, and (iii) optimal rate convergence. A few numerical results are also presented in this article, which demonstrates the robustness of the proposed numerical scheme.

中文翻译：

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Poisson-Nernst-Planck 系统的一种保正性、能量稳定和收敛的数值方案

在本文中，我们提出并分析了 Poisson-Nernst-Planck 方程 (PNP) 系统的有限差分数值方案。为了理解 PNP 模型的能量结构，我们使用了能量变分法 (EnVarA)，这样 PNP 系统就可以重新表述为非常量流动性 $H^{-1}$ 梯度流，具有奇异对数涉及的能量势。为了确保唯一的可解性和能量稳定性，迁移率函数被明确处理，而对数和电势扩散项都被隐式处理，这是由于这两个能量函数部分的凸性质。两种浓度 $n$ 和 $p$ 的阳性保留特性是在理论水平上建立的。这是基于一个微妙的事实，即围绕 $0$ 值的对数项的奇异性质阻止数值解达到奇异值，因此数值方案始终是明确定义的。此外，在这项工作中提供了最佳速率收敛分析，由于非线性抛物线系数，其中必须涉及许多高度非标准的估计。更高阶的渐近扩展（高达三阶时间精度和四阶空间精度），粗略的误差估计（建立 $n$ 和 $p$ 的 $\ell^\infty$ 界限），以及细化误差估计必须执行才能实现这样的收敛结果。据我们所知，这项工作将是第一个结合以下三个理论特性的 PNP 系统数值方案：(i) 独特的可解性和正性，(ii) 能量稳定性，以及 (iii) 最佳速率收敛。本文还提供了一些数值结果，证明了所提出的数值方案的鲁棒性。