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.

中文翻译：

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Poisson-Nernst-Planck 方程的一种完全离散的正性保持和能量耗散有限差分格式

Poisson-Nernst-Planck (PNP) 方程是一种宏观模型，广泛用于描述离子通道中离子传输的动力学。在本文中，我们为有界域中的 PNP 方程引入了一种半隐式有限差分格式。考虑了泊松方程的一般边界条件。显示完全离散的方案满足以下属性：质量守恒、无条件正性和能量耗散（因此保持稳态）。证明了半离散方案的可解性，并提出了一个简单的不动点迭代来求解完全离散方案。提供了 1D 和 2D 以及多个物种的数值示例，以证明所提出方案的收敛性和特性。