离散Riesz空间分数阶对流-扩散方程中线性方程组的τ矩阵预处理方法  

A τ MATRIX BASED PRECONDITIONING METHOD FOR SOLVING THE SYSTEM OF LINEAR EQUATIONS FROM THE DISCRETIZATION OF THE RIESZ SPACE FRACTIONAL ADVECTION-DISPERSION EQUATIONS

作　　者：唐世平 黄玉梅[1] Tang Shiping;Huang Yumei(School of Mathematics and Statistics,Lanzhou University,Lanzhou 730000,China)

机构地区：[1]兰州大学数学与统计学院,兰州730000

出　　处：《计算数学》2023年第4期483-496,共14页Mathematica Numerica Sinica

基　　金：国家自然科学基金(11971215,11971214);兰州大学应用数学与复杂系统重点实验室资助。

摘　　要：在Riesz空间分数阶对流-扩散方程的数值求解中,通过采用加权移位的Grünwald差分格式对其空间导数进行离散以及Crank-Nicolson格式对其时间导数进行离散,得到一个系数矩阵为单位矩阵与两个对称正定Toeplitz矩阵之和的线性方程组.在本文中,对该线性方程组,利用其系数矩阵的结构,提出了一种τ预处理矩阵,并采用预处理共轭梯度法求解了该线性方程组.理论分析给出了预处理后系数矩阵的谱分布以及条件数估计.数值实验结果也说明了所构造的预处理矩阵在采用预处理共轭梯度法求解Riesz空间分数阶对流-扩散方程离散后得到的线性方程组的有效性.In this paper,the numerical methods for the Riesz space fractional convection-dispersion equation is considered.By applying the weighted shifted Grünwald difference scheme to discretize the spatial fractional derivatives and the Crank-Nicolson difference scheme to discretize the temporal derivative in the Riesz space fractional convection-dispersion equation,respectively,we get the discrete results as a system of linear equations whose coefficient matrix is the sum of an identity matrix and two symmetric positive definite Toeplitz matrices.A τ preconditioner is constructed and the preconditioned conjugate gradient method is applied to solve the discrete system of linear equations.The spectral distribution of the preconditioned matrix is analyzed and the condition number of the preconditioned matrix is estimated.Numerical experiments show that the constructed preconditioner is very effective when it is combined in the preconditioned conjugate gradient method to solve the discrete system of linear equations.

关 键 词：Riesz空间分数阶对流-扩散方程 Crank-Nicolson有限差分格式 条件数 τ预处理矩阵 谱分析 

分 类 号：O241.82[理学—计算数学]