二维分数阶发展型方程的正式的二阶BDF交替方向隐式紧致差分格式  

A Formally Second-Order BDF Compact ADI Difference Scheme for the Two-Dimensional Fractional Evolution Equation

作　　者：陈红斌 甘四清[1] 徐大[3] 彭玉龙[2] 

机构地区：[1]中南大学数学与统计学院,长沙410083 [2]中南林业科技大学理学院,长沙410004 [3]湖南师范大学数学与计算机科学学院,高性能计算与随机信息处理省部共建教育部重点实验室,长沙410081

出　　处：《数学物理学报（A辑）》2017年第5期976-992,共17页Acta Mathematica Scientia

基　　金：国家自然科学基金(11571373;11671131);湖南省重点学科建设项目~~

摘　　要：该文将研究二维分数阶发展型方程的正式的二阶向后微分公式(BDF)的交替方向隐式(ADI)紧致差分格式.在时间方向上用二阶向后微分公式离散一阶时间导数,积分项用二阶卷积求积公式近似,在空间方向上用四阶精度的紧致差分离散二阶空间导数得到全离散紧致差分格式.基于与卷积求积相对应的实二次型的非负性,利用能量方法研究了差分格式的稳定性和收敛性,理论结果表明紧致差分格式的收敛阶为O(k^(a+1)+h_1~4+h_2~4),其中k为时间步长,h_1和h_2分别是空间x和y方向的步长.最后,数值算例验证了理论分析的正确性.In this paper, we will consider a formally second-order backward differentiation formula （BDF） compact alternating direction implicit （ADI） difference scheme for the two- dimensional fractional evolution equation. To obtain a fully discrete implicit scheme, the inte- gral term is treated by means of the second order convolution quadrature suggested by Lubich and the second order space derivatives are approximated by the fourth-order accuracy compact finite difference. The stability and convergence of the compact difference scheme in a new norm are proved by the energy method. The verification of stability and convergence is based on the nonnegative character of the real quadratic form associated with the convolution quadrature. A numerical experiment in total agreement with our analysis is reported.

关 键 词：二维分数阶发展型方程 二阶BDF ADI 紧致差分格式 稳定性 收敛性 

分 类 号：O241.82[理学—计算数学]