分数阶Nernst-Planck方程的有限差分/谱元法求解  

A Finite Difference/Spectral Element Method for the Fractional Nernst-Planck Equation

作　　者：李娴娟[1] 许传炬[1] 

机构地区：[1]厦门大学数学科学学院,厦门361005

出　　处：《工程数学学报》2010年第2期207-218,共12页Chinese Journal of Engineering Mathematics

基　　金：国家自然科学基金(10531080);973"高性能科学计算研究"项目(2005CB321703);福建省自然科学基金(S0750017)~~

摘　　要：Nernst-Planck方程是用来描述在离子浓度梯度C及电场V共同存在的情况下,穿过渗透膜的离子(如钙,钾,钠,氯,镁等)流J的方程。但是,计算Nernst-Planck方程的数值解会遇到一些困难。本文考虑用以描述神经细胞中离子反常扩散现象的电缆型简化的分数阶Nernst-Planck方程,提出了一个时间有限差分/空间谱元法对该方程进行数值求解。我们给出了数值方法的详细构造过程以及实现方法。结果表明数值解在空间方向上具有指数阶收敛精度,在时间方向上具有2?α阶精度。最后,通过计算一个具有实际背景参数的问题说明所提方法的潜在应用。The Nernst-Planck equation describes the flux of ions （for example, calcium, potassium, sodium, chloride, and magnesium etc.） through a diffusive membrane under the influence of both the ionic concentration gradient VC and electric potential VV. However, numerical approximations to the Nernst-Planck equation suffer from several difficulties. In this paper, we first briefly recall the derivation of the fractional Nernst-Planck equation in a cable-like geometry, which describes the anomalous diffusion in the movement of the ions in a neuronal system. Then a method combining finite differences in time and spectral element methods in space is proposed to numerically solve the underlying problem. The detailed construction and implementation of the method are presented. Our numerical experiences show that the convergence of the proposed method is exponential in space and （2 - a）-order in time. Finally, a practical problem with realistic physical parameters is simulated to demonstrate the potential applicability of the method.

关 键 词：分数阶Nernst—Planck方程:谱元法 有限差分法 Newton.Krylov迭代法 

分 类 号：O241.8[理学—计算数学]