Several Kinds Of Structure-Preserve Preconditioners For Time-space Fractional Diffusion Equations

Fractional diffusion equations is a kind of diffusion equations with fractional derivatives.In some applications,fractional diffusion equations can more accurately and realistically reflect the variation process of complex systems than traditional integer diffusion equations.However the nonlocality of fractional diffusion operators caused the coefficient matrix of discrete algebraic equations is often dense,which brings great diffculties to numerical solution.Therefore,it is necessary to study the fast solution method of fractional diffusion equations.This paper studies the time-space one-dimensional and two-dimensional fractional diffusion equations,and the main work is to solve the preconditioning method with time fractional diffusion equations.After the finite difference discretization of a class of equations we studied,we construct a preconditioned iterative algorithm by means of Kronecker produc,the properties of Toeplitz matrix and circulant matrix.The first chapter mainly introduces fractional order equations and some basic knowledge that the paper involved.The second chapter studies the time-space one-dimensional fractional diffusion equations.We perform finite difference discretization in both time and space directions of this kind of equations.And the obtained linear system is solved in the fully coupled time framework.The coefficient matrix of the obtained linear system is large and dense,and the computional cost is relatively large,so we construct a fast preconditioning strategy based on Kronecker product,block diagonal,and block triangle.All preconditioning methods use structure-preserve methods to approximate discrete spatial fractional diffusion operators.Numerical experiments show that when the time fractional derivative is close to the first order,the efficiency of four kinds of block preconditioning is better,and faster than the traditional time advancement method.The third chapter studies the time-space two-dimensional variable coefficient fractional diffusion equations,we performs finite difference discretization in the time-space direction and study the preconditioning method of the discrete linearsystem.The linear system generated after discretization is represented by the sum of five Kronecker products.For this structure,we use a Kronecker product to approximate the sum of five Kronecker products,we use the structure retention methord to approximate the fractional diffusion discrete matix in this Kronecker product.Numerical examples illustrate the effectiveness of this approach.