二维分数阶发展型方程交替方向隐式紧致差分格式  

Alternating direction implicit compact difference scheme for the two-dimensional fractional evolution equation

作　　者：黎丽梅[1] 张书华[2] 彭龙[3] 

机构地区：[1]湖南理工学院数学学院,岳阳414006 [2]天津财经大学数学经济研究中心,天津300222 [3]北京外国语大学商学院,北京100089

出　　处：《中国科学：数学》2015年第8期1265-1280,共16页Scientia Sinica：Mathematica

基　　金：国家自然科学基金(批准号:11171251)资助项目

摘　　要：本文考虑用交替方向隐式(ADI)方法研究二维分数阶发展型方程(带有弱奇异核的积分-微分方程)的数值解,在空间方向上使用紧致差分,时间方向上采用Crank-Nicolson格式,积分项用二阶卷积求积公式逼近.此外,本文还给出全离散格式,并利用离散的能量法证明全离散格式是无条件稳定和收敛的,且收敛阶为O(Υ~2+h_x^4+h_y^4),其中7是时间步长,h_x和h_y分别是空间x和y方向的步长.最后,本文用数值例子验证理论分析的正确性.In this paper, we consider the numerical solutions of the two dimensional fractional evolution equa- tion （integro-differential equation with a weakly singular kernel） with alternating direction implicit （ADI） method. The compact difference approach is used for the spatial discretization, and, for the time stepping, the Crank- Nicolson scheme combined with the second order convolution quadrature approximating the integral term is considered. The unconditional stability and L2 norm convergence of the scheme are proved rigorously. The con- vergence order is O（τ2＋h4x＋h4y）, where T is the temporal grid size and hx, hy are spatial grid sizes in the x and y directions, respectively. Numerical results are presented to support our theoretical analysis.

关 键 词：发展型方程 ADI格式 紧致差分 离散的能量方法 弱奇异核 

分 类 号：O241.82[理学—计算数学]