A positivity-preserving, energy stable and convergent numerical scheme for the Poisson-Nernst-Planck system
A positivity-preserving, energy stable and convergent numerical scheme for the Poisson-Nernst-Planck system
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DOI：
10.1090/mcom/3642
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发表时间：
2020-09
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影响因子：
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通讯作者：
Chun Liu;Cheng Wang;S. Wise;Xingye Yue;Shenggao Zhou
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作者：
Chun Liu;Cheng Wang;S. Wise;Xingye Yue;Shenggao Zhou
本文提出并分析了Poisson-Nernst-Planck方程（PNP）的一种有限差分数值格式。为了理解PNP模型的能量结构，我们利用能量变分方法（EnVarA），使PNP系统可以重新表述为一个非恒定迁移率H − 1 H^{-1}梯度流，其中包含奇异对数能量势。为了确保唯一的可解性和能量稳定性，显式处理迁移率函数，而对数和电势扩散项被隐式处理，由于这两个能量泛函部分的凸性质。在理论水平上建立了两种浓度，n n和p p的正性保持属性。这是基于一个微妙的事实，即0 - 0值周围的对数项的奇异性防止数值解达到奇异值，因此数值方案总是定义良好的。此外，在这项工作中，提供了一个最佳的速度收敛分析，其中许多高度非标准的估计，由于非线性抛物系数。高阶渐近展开（时间精度达到三阶，空间精度达到四阶），粗误差估计（建立n n和p p的n ∞ \ell ^\infty界），以及精误差估计都必须进行，以达到这样的收敛结果.据我们所知，这项工作将是第一个联合收割机结合以下三个理论性质的数值方案的PNP系统：（i）唯一的可解性和积极性，（ii）能量稳定性，和（iii）最佳速率收敛。本文还给出了一些数值结果，这表明所提出的数值格式的鲁棒性。
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 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 n and p 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 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 ℓ ∞ \ell ^\infty bound for n n and p 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.