A second-order accurate numerical approximation for the fractional diffusion equation
A second-order accurate numerical approximation for the fractional diffusion equation
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DOI：
10.1016/j.jcp.2005.08.008
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发表时间：
2006-03-20
影响因子：
4.1
通讯作者：
Scheffler, HP
中科院分区：
文献类型：
--
作者：
Tadjeran, C;Meerschaert, MM;Scheffler, HP
分数阶扩散方程是经典扩散方程的推广，用于处理超扩散过程。本文研究了在有限域上求解一类变系数初边值分数阶扩散方程的一种实用的二阶精确时间和空间数值方法。在经典的Crank-Nicholson方法的基础上，结合空间外推法获得了时间和空间上的二阶精确数值估计。检验了该方法的稳定性、一致性和（因此）收敛性。证明了基于移位Grunwald公式的分数阶Crank-Nicholson方法是无条件稳定的。给出了一个数值算例，并与精确解析解的收敛阶进行了比较。(c) 2005爱思唯尔公司版权所有。
Fractional order diffusion equations are generalizations of classical diffusion equations, treating super-diffusive flow processes. In this paper, we examine a practical numerical method which is second-order accurate in time and in space to solve a class of initial-boundary value fractional diffusive equations with variable coefficients on a finite domain. An approach based on the classical Crank-Nicholson method combined with spatial extrapolation is used to obtain temporally and spatially second-order accurate numerical estimates. Stability, consistency, and (therefore) convergence of the method are examined. It is shown that the fractional Crank-Nicholson method based on the shifted Grunwald formula is unconditionally stable. A numerical example is presented and compared with the exact analytical solution for its order of convergence. (c) 2005 Elsevier Inc. All rights reserved.