A fully discrete positivity-preserving and energy-dissipative finite difference scheme for Poisson-Nernst-Planck equations
A fully discrete positivity-preserving and energy-dissipative finite difference scheme for Poisson-Nernst-Planck equations
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DOI：
10.1007/s00211-020-01109-z
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发表时间：
2020-05-01
影响因子：
2.1
通讯作者：
Huang, Xiaodong
中科院分区：
文献类型：
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作者：
Hu, Jingwei;Huang, Xiaodong
泊松-能斯特-普朗克(Poisson-Nernst-Planck，PNP)方程是一个广泛用于描述离子通道中离子输运动力学的宏观模型。本文介绍了一种求解有界域上PNP方程的半隐式有限差分格式。考虑了泊松方程的一般边界条件。全离散格式满足以下性质：质量守恒、无条件正性和能量耗散(从而保持定态)。证明了半离散格式的可解性，并提出了求解全离散格式的简单不动点迭代法。文中给出了一维、二维和多物种情况下的数值算例，证明了该格式的收敛和性质。
The Poisson-Nernst-Planck (PNP) equations is a macroscopic model widely used to describe the dynamics of ion transport in ion channels. In this paper, we introduce a semi-implicit finite difference scheme for the PNP equations in a bounded domain. A general boundary condition for the Poisson equation is considered. The fully discrete scheme is shown to satisfy the following properties: mass conservation, unconditional positivity, and energy dissipation (hence preserves the steady state). Solvability of the semi-discrete scheme is proved and a simple fixed point iteration is proposed to solve the fully discrete scheme. Numerical examples in both 1D and 2D and for multiple species are presented to demonstrate the convergence and properties of the proposed scheme.