Poisson-Nernst-Planck方程的数值方法及其应用

基于平均场近似的Poisson-Nernst-Planck（PNP）方程是离子通道中离子输运的模型。为了获得物理上可靠的数值解，在离散水平上保持解析解的物理性质，如质量守恒性、能量耗散性和浓度的正性是非常必要和关键的。在对流占优情况下，标准的PNP数值模拟会导致数值振荡，并无法保证PNP方程的物理性质。针对以上问题，本项目提出在对流占优情况下，发展稳健的保物理性质的PNP类型数值格式，并设计相应的数值算法应用至离子通道中离子的输运。研究内容包括：（1）发展PNP方程结合密度泛函理论（DFT）的PNP-DFT数值格式，来满足PNP方程的物理性质；（2）在对流占优下，发展稳健的PNP类型数值格式，来满足PNP方程的物理性质；证明高维对流占优的Nernst-Planck类型数值格式能退化到的迎风格式。（3）将PNP类型数值方法应用至离子通道中的离子输运，并将数值结果与其他数值模拟结果进行对比。

Poisson-Nernst-Planck (PNP) equations based on the mean-field approximation is a model of ionic transport in ion channels. In order to obtain physically reliable numerical solutions, it is very necessary and critical to preserve physical properties of analytical solutions at the discrete level, such as mass conservation, energy dissipation and positivity of concentrations. In the convection dominant regime, the standard PNP numerical simulation will cause numerical oscillations, and can not preserve the physical properties of PNP equations, so that it can not be applied to the ionic transport in ion channels. In response to the above problems, this project proposes to develop a robust PNP-type numerical scheme to preserve physical properties in the convection dominant regime, and designs corresponding numerical algorithms to apply to the ionic transport in ion channels. The contents of the research include: (1) Developing a PNP-DFT numerical scheme of PNP equations combined with density functional theory (DFT) to satisfy physical properties of PNP equations; (2) Developing a robust PNP-type numerical scheme to satisfy physical properties of PNP equations in the convection dominant regime; Proving that the high-dimensional, convection-dominated Nernst-Planck-type numerical scheme can degenerate to the upwind scheme. (3) Applying the numerical methods of PNP-type equations to the ionic transport in ion channels, and the numerical results are compared with other results of numerical simulations.