求解三维空间分数阶对流扩散方程的Douglas-Gunn格式  

Douglas-Gunn Finite Difference Scheme for Three-dimensional Space Fractional Advection Diffusion Equation

作　　者：聂玉峰[1] 胡嘉卉 王俊刚[1] NIE Yufeng;HU Jiahui;WANG Jungang(Research Center for Computational Science,Northwestern Polytechnical University,Xi′an 710129,China;College of Science,Henan University of Technology,Zhengzhou 450001,China)

机构地区：[1]西北工业大学计算科学研究中心,陕西西安710129 [2]河南工业大学理学院,河南郑州450001

出　　处：《郑州大学学报（理学版）》2019年第1期44-50,共7页Journal of Zhengzhou University:Natural Science Edition

基　　金：国家自然科学基金项目(11471262)

摘　　要：由于分数阶导数的非局部性特征,在模拟反常扩散现象时使用分数阶偏微分方程具有更好的效果,但是分数阶导数的非局部性也给数值分析和计算带来了很大困难,尤其在多维空间情形下.通过对经典Douglas-Gunn格式的推广,提出一种求解三维空间分数阶对流扩散方程(space fractional advection diffusion equation,SFADE)的交替方向隐(alternating direction implicit,ADI)差分格式,并用矩阵法证明了其稳定性和收敛性.用数值算例进一步验证了该格式在空间和时间方向均具有较高的二阶收敛精度,可以高效地求解三维SFADE.Due to the non-locality of fractional derivatives,fractional partial differential equations were better to describe anomalous diffusion phenomena than other methods.However,while enjoying the convenience from mathematical modeling,it also caused lots of trouble especially in solving multidimensional cases.An efficient numerical algorithm was proposed for solving the three-dimensional space fractional advection diffusion equation(SFADE)by generalizing the Douglas-Gunn scheme.Stability and convergence of the method were proved by the matrix method.The derived alternating direction implicit(ADI)finite difference scheme had the second order accuracy in both time and space directions,respectively.The efficiency and convergence orders were finally demonstrated by some numerical examples.

关 键 词：三维SFADE ADI格式 CRANK-NICOLSON格式 Douglas-Gunn格式 稳定性 收敛性 

分 类 号：O241[理学—计算数学]