保正的Poisson-Nernst-Planck方程的有限差分格式

Positivity-preserving finite difference schemes for the Poisson-Nernst-Planck equations

提出一种简单有效的有限差分方法求解含有多种离子的二维Poisson-Nernst-Planck(PNP)方程。基于调和平均近似,对Nernst-Planck(NP)方程提出新的中心差分离散。在时间上使用向后欧拉离散导出线性化半隐格式。对于时间步长没有任何约束,数值分析可以证明该格式遵循离子质量守恒性和离子浓度正性。此外,对半隐式NP格式的条件数进行理论与计算研究,进一步揭示该格式在计算效率和稳定性方面的优势。

We develop simple but effective finite difference methods for solving the two-dimensional Poisson-Nernst-Planck(PNP)equations with multiple ionic species.A novel central-differencing discretization based on the harmonic-mean approximations is proposed for the Nernst-Planck(NP)equations.The backward euler discretizations in time is employed to derive a linearized semi-implicit scheme.Without any restriction for the time step size,numerical analysis proves that the numerical scheme respect:ionic mass conservative and positivity-preserving.In addition,theoretical and computational invertigations are performed to study condition numbers of the semi-implicit discretization of the NP equations,further revealing advantages of the scheme in computational efficiency and stability.