Convergence Analysis of Structure-Preserving Numerical Methods Based on Slotboom Transformation for the Poisson-Nernst-Planck Equations
Convergence Analysis of Structure-Preserving Numerical Methods Based on Slotboom Transformation for the Poisson-Nernst-Planck Equations
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DOI：
10.4310/cms.2023.v21.n2.a7
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发表时间：
2022-02
期刊：
影响因子：
--
通讯作者：
Jie Ding;Cheng Wang;Shenggao Zhou
中科院分区：
文献类型：
--
作者：
Jie Ding;Cheng Wang;Shenggao Zhou
Poisson-Nernst-Planck（PNP）方程组的保结构数值方法是近年来研究的热点。在这项工作中，我们提供了一个最佳的速度收敛性分析和误差估计有限差分格式的基础上Slotboom重新制定。在有限差分空间离散化中，考虑了交错网格点处的不同流动性平均值，例如调和平均值、几何平均值、算术平均值和熵平均值。一个半隐式的时间离散化，这反过来又导致在每个时间步长的一个非常数系数，正定线性系统。在相容性分析中应用了高阶渐近展开式，这种高阶相容性估计对于控制浓度变量的离散最大范数是必要的。在收敛性估计中，为了简化计算，采用了迁移率平均的调和平均，这给理论分析带来了很大的方便，而其他迁移率平均的选择也会导致期望的误差估计，涉及更多的技术细节。其结果是，浓度，电势和离子通量的最优速率收敛分析，这是第一个这样的结果的结构保持数值格式的基础上Slotboom重新制定。值得注意的是，收敛性分析为条件能量耗散分析提供了理论依据，该分析依赖于浓度和电位梯度的最大范数界。一些数值结果也证明了相关的格式的精度和结构保持性能。
The analysis of structure-preserving numerical methods for the Poisson--Nernst--Planck (PNP) system has attracted growing interests in recent years. In this work, we provide an optimal rate convergence analysis and error estimate for finite difference schemes based on the Slotboom reformulation. Different options of mobility average at the staggered mesh points are considered in the finite-difference spatial discretization, such as the harmonic mean, geometric mean, arithmetic mean, and entropic mean. A semi-implicit temporal discretization is applied, which in turn results in a non-constant coefficient, positive-definite linear system at each time step. A higher order asymptotic expansion is applied in the consistency analysis, and such a higher order consistency estimate is necessary to control the discrete maximum norm of the concentration variables. In convergence estimate, the harmonic mean for the mobility average, which turns out to bring lots of convenience in the theoretical analysis, is taken for simplicity, while other options of mobility average would also lead to the desired error estimate, with more technical details involved. As a result, an optimal rate convergence analysis on concentrations, electric potential, and ionic fluxes is derived, which is the first such results for the structure-preserving numerical schemes based on the Slotboom reformulation. It is remarked that the convergence analysis leads to a theoretical justification of the conditional energy dissipation analysis, which relies on the maximum norm bounds of the concentration and the gradient of the electric potential. Some numerical results are also presented to demonstrate the accuracy and structure-preserving performance of the associated schemes.