Riesz空间分数阶对流-扩散方程的一种新型Crank-Nicolson有限体积法  

A Novel Crank-Nicolson Finite Volume Method for Riesz Space Fractional Advection-diffusion Equations

作　　者：屈威 王庆勇 QU WEI;WANG QINGYONG(School of Mathematics and Statistics,Shaoguan University,Shaoguan 512005,China;Science and Technology on Information Systems Engineering Laboratory,National University of Defense Technology,Changsha 410073,China)

机构地区：[1]韶关学院数学与统计学院,韶关512005 [2]国防科技大学信息系统工程重点实验室,长沙410073

出　　处：《应用数学学报》2024年第3期402-416,共15页Acta Mathematicae Applicatae Sinica

基　　金：国家自然科学基金(批准号:62301006);广东省基础与应用基础研究基金(批准号:2020A1515110454)资助项目。

摘　　要：分数阶微分方程作为整数阶微分方程的推广,近年来被广泛应用于科学和工程领域,从而受到越来越多学者的关注.本文提出一种新型Crank-Nicolson有限体积方法求解具有Dirichlet齐次边界的Riesz空间分数阶对流-扩散方程.为了得到Riesz空间分数阶对流-扩散方程的离散格式,在时间层上,利用Crank-Nicolson方法对一阶时间偏导数进行离散.在空间层上,利用有限体积法近似对流项的一阶空间偏导数和扩散项的Riesz空间分数阶偏导数.更进一步,我们也得到了该Crank-Nicolson有限体积离散格式的稳定性和收敛性两个主要理论结果.证明了该离散格式是无条件稳定的,以及在离散L2-范数下的收敛阶为O(h2+τ2),其中h和τ分别为空间和时间上的步长.最后,通过数值试验验证了该离散格式理论结果的正确性.As a generalization of integer-order differential equations,fractional dif-ferential equations have been widely used in science and engineering in recent years,which also have attracted widespread attention from many scholars.In this paper,a novel Crank-Nicolson finite volume method(CN-FVM)is proposed for solving Riesz space-fractional advection-diffusion equations(RSFADEs)with homogeneous Dirichlet boundary conditions.In order to obtain the discrete linear systems arising from RS-FADEs,the Crank-Nicolson method is used to discretize the first order time partial derivative,while the finite volume method is adopted to approximate the first order space partial derivative of advection term and the Riesz space fractional partial deriva-tive of the diffusion term.Furthermore,two main theoretical results about stability and convergence of the CN-FVM scheme are also discussed.It is proved that CN-FVM scheme is unconditionally stable and convergent with the accuracy of O(h?++2)in the discrete L2-norm,where h and τ denote the spatial and temporal step sizes,respec-tively.Finally,some numerical experiments are presented to confirm the correctness of the theoretical analysis of the proposed scheme.

关 键 词：Riesz空间分数阶对流-扩散方程 Crank-Nicolson方法 有限体积法 无条件稳定性 收敛性 离散L2-范数 

分 类 号：O241.82[理学—计算数学]