论文标题
基于泊松 - nernst-planck方程的槽脉转换的结构保存数值方法的收敛分析
Convergence Analysis of Structure-Preserving Numerical Methods Based on Slotboom Transformation for the Poisson--Nernst--Planck Equations
论文作者
论文摘要
近年来，对Poisson-Nernst-Planck（PNP）系统的结构保存数值方法的分析吸引了日益增长的兴趣。在这项工作中，我们基于Slotboom重新制定提供了有限差异方案的最佳速率收敛分析和错误估计。在有限差异空间离散化中考虑了交错的网格点的移动性平均值，例如谐波平均值，几何平均值，算术平均值和熵均值。应用了半幅度的时间离散化，这又导致每个时间步长的非恒定系数，正定线性系统。在一致性分析中应用了高阶渐近扩展，并且必须进行高阶一致性估计来控制浓度变量的离散最大规范。在收敛估计中，为简单起见，在理论分析中带来了很多便利性的谐波平均值，这在理论分析中带来了很多便利，而其他流动性的其他选择也将导致所需的错误估计值，涉及更多技术细节。结果，得出了对浓度，电势和离子通量的最佳速率收敛分析，这是基于Slotboom重新印象的结构保存数值方案的第一个结果。有人指出，收敛分析导致条件能量耗散分析的理论理由，这取决于浓度的最大规范界限和电势的梯度。还提出了一些数值结果，以证明相关方案的准确性和结构性能。
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.
