发表机构
University of Electronic Science and Technology of China; Southern University of Science and Technology (SUSTech)（电子科技大学; 南方科技大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对PNP方程在小介电常数和复杂边界下的数值挑战，基于其瓦瑟斯坦梯度流结构构建带泊松约束的JKO格式，开发高效原始-对偶算法，经数值实验验证其保结构特性与计算效率。
AI 中文摘要
泊松-能斯特-普朗克（PNP）方程通过泊松方程将离子输运与静电相互作用强耦合，在小介电常数和复杂势边界条件下带来了重大数值挑战。这些方程的基础是自然的瓦瑟斯坦梯度流结构，其中泊松方程作为非局域静电相互作用能的局部实现。利用该结构，我们将每个时间步构建为一个带约束的凸极小化问题，将离子连续性方程和泊松方程作为线性约束纳入，使得浓度、通量和静电势能够同时更新。该格式的变分结构在一般静电边界条件下固有地保证了原始自由能的耗散、质量守恒以及离子浓度的非负性。此外，该框架在结构上是模块化的：从经典PNP模型扩展到具有空间位阻相互作用和浓度梯度修正的改进PNP模型，仅需修改能量泛函，而所有保结构性质会自动保留。为了高效求解所得的大规模约束问题，我们开发了预处理和变换后的原始-对偶算法，配备了定制的快速对偶求解器，即基于离散余弦变换（DCT）的直接方法和舒尔补迭代方法，这些方法利用了PDE约束的耦合块结构。对经典和改进PNP系统的数值实验证明了该格式的准确性和保结构性质，且表明所提出的算法在强耦合小介电常数区域中可靠收敛，而计算成本无显著增长。
英文摘要
The Poisson--Nernst--Planck (PNP) equations strongly couple ionic transport and electrostatic interactions through the Poisson equation, posing substantial numerical challenges under small permittivity and complex potential boundary conditions. Underlying these equations is a natural Wasserstein gradient-flow structure, in which the Poisson equation serves as a local realization of the nonlocal electrostatic interaction energy. Exploiting this structure, we formulate each time step as a constrained convex minimization problem where the ionic continuity equations and the Poisson equation are incorporated as linear constraints, allowing the concentrations, fluxes, and electrostatic potential to be updated simultaneously. The variational structure of the scheme intrinsically guarantees the dissipation of the original free energy, mass conservation, and nonnegativity of ionic concentrations under general electrostatic boundary conditions. Moreover, the framework is structurally modular: extending from classical to modified PNP models with steric interactions and concentration-gradient corrections requires only modifying the energy functional, while all structure-preserving properties are automatically retained. To efficiently solve the resulting large-scale constrained problems, we develop preconditioned and transformed primal--dual algorithms equipped with tailored fast dual solvers, namely DCT-based direct and Schur-complement iterative methods, that exploit the coupled block structure of the PDE constraints. Numerical experiments on classical and modified PNP systems demonstrate the accuracy and structure-preserving properties of the scheme, and show that the proposed algorithms converge reliably in strongly coupled small-permittivity regimes without significant growth in computational cost.
