Kronecker Product Based Preconditioned Iterative Algorithms For High-dimensional Space-fractional Diffusion Equations

Fractional diffusion equations has found increasingly extensive applications to different fields and subjects,such as physics and biology.Since fractional diffusion equations seldom has a closed-form solution,it is of vital importance to study the numerical solutions.So far,a variety of methods,including spectral method,finite element method and finite difference method,have emerged for research into the numerical solutions.However,most of them,while working out the numerical solutions,cannot avoid generating a dense coefficient matrix,which will consume more computational cost and occupy more memory.Considering the above problem,this paper conducts Kronecker product splitting iteration of the discrete linear system based on the properties of Kronecker product to accelerate Krylov subspace.This method can solve the two-dimensional and three-dimensional space-fractional diffusion equations.Apart from reducing the computing complexity and the iteration times during the iteration process,it also reduces the computing time and occupation of internal storage.Chapter 1 first introduces the fractional diffusion equations,and then points out the background and development of the fractional differential equation numerical methods.Following that,the knowledge and background of Krylov subspace acceleration algorithm and preconditioning methods are introduced.Finally,the main research work of this paper is introduced.Chapter 2 first provides the definition of several fractional derivatives required by this paper,such as the fractional derivative defined by Caputo,Riemann-Liouville and Grünwald-Letnikov,respectively.Next,Kronecker product's operation and its properties as well as Toeplitz matrix's computing and approximation used in this paper is expounded,including the product of Toeplitz matrix and vector.Chapter 3 studies the numerical solution of the two-dimensional fractional diffusion equation.To start with,we use the finite difference method to discrete it in space and time.Then,the focus is on introducing how to accelerate the process of the discrete linear system using the preconditioning method based on one-parameter and two-parameter Kronecker product splitting iteration.Meanwhile,the convergence of Kronecker product splitting iteration is analyzed,and the optimal parameter for the least iteration times is given.Finally,numerical experiments are conducted to verifycorrectness and validity of this method.Chapter 4 applies this preconditioning method to the three-dimensional fractional diffusion equation,and provides the specific operation process.The convergence of Kronecker product splitting iteration is analyzed,and the three-dimensional numerical solution example is provided to verify this method's correctness and validity via experiments.Chapter 5 summarizes the findings of this paper,and looks into the future research directions of this field.