SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 2004-2024, August 2024.
Abstract. We propose and analyze a novel approach to construct structure preserving approximations for the Poisson–Nernst–Planck equations, focusing on the positivity preserving and mass conservation properties. The strategy consists of a standard time marching step with a projection (or correction) step to satisfy the desired physical constraints (positivity and mass conservation). Based on the [math] projection, we construct a second order Crank–Nicolson type finite difference scheme, which is linear (exclude the very efficient [math] projection part), positivity preserving, and mass conserving. Rigorous error estimates in the [math] norm are established, which are both second order accurate in space and time. The other choice of projection, e.g., [math] projection, is discussed. Numerical examples are presented to verify the theoretical results and demonstrate the efficiency of the proposed method.

中文翻译：

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泊松-能斯特-普朗克方程的保正性和质量守恒投影法


SIAM 数值分析杂志，第 62 卷，第 4 期，2004-2024 页，2024 年 8 月。

抽象的。我们提出并分析了一种为泊松-能斯特-普朗克方程构造结构保持近似的新方法，重点关注正性保持和质量守恒性质。该策略由标准时间推进步骤和投影（或校正）步骤组成，以满足所需的物理约束（正性和质量守恒）。基于[数学]投影，我们构造了二阶Crank-Nicolson型有限差分格式，该格式是线性的（排除非常有效的[数学]投影部分）、正性保持和质量守恒。建立了[数学]范数中严格的误差估计，其在空间和时间上都是二阶准确的。讨论了投影的其他选择，例如[数学]投影。数值例子验证了理论结果并证明了该方法的有效性。