We develop a set of numerical schemes for the Poisson–Nernst–Planck equations. We prove that our schemes are mass conservative, uniquely solvable and keep positivity unconditionally. Furthermore, the first-order scheme is proven to be unconditionally energy dissipative. These properties hold for various spatial discretizations. Numerical results are presented to validate these properties. Moreover, numerical results indicate that the second-order scheme is also energy dissipative, and both the first- and the second-order scheme preserves the maximum principle for cases where the equation satisfies the maximum principle.

中文翻译：

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Poisson-Nernst-Planck 方程的无条件保正性和能量耗散方案

我们为Poisson-Nernst-Planck方程开发了一套数值方案。我们证明我们的方案是大规模保守的，唯一可解的，并且无条件地保持积极性。此外，一阶方案被证明是无条件耗能的。这些属性适用于各种空间离散化。提供了数值结果来验证这些特性。此外，数值结果表明，二阶格式也是耗能的，对于方程满足最大值原理的情况，一阶和二阶格式都保留了最大值原理。