Matrix Splitting Iteration Method For Space Fractional Diffusion Equations

Due to its genetic and memory characteristics,the fractional diffusion equation has been widely used in different fields of science and engineering,such as Newtonian fluid mechanics,turbulence,and abnormal diffusion phenomena.Therefore,the study of the fractional diffusion equation has certain scientific value.Research has shown that its analytical solution is not easy to obtain,so it can only be found for its numerical solution.One of the main characteristics of fractional differential operators is that it is non-local,which makes the simple discretization scheme of the fractional diffusion equation unconditionally unstable even if it is implicit.Recently,based on the implicit finite-difference scheme with displacement Gr¨unwald formula.the fractional diffusion equation is unconditionally stable,and its coefficient matrix is the sum of the identity matrix with the diagonal-times-Toeplitz matrices.This paper proposes the ADI-like iteration method and the Hessenberg matrix splitting iteration method for this type of linear system,and gives the corresponding preconditioners,theoretical analysis and numerical experiments have compared the convergence and convergence speed of different iteration methods.The main research work is as follows:1.For the numerical discretization form of the space fractional diffusion equation,the derivative in time can be discretized by the standard first-order time difference quotient,and the fractional derivative in space can be approximated by Gr¨unwald Formula,the discretization linear system has a Toeplitz-like matrix structure,which can be written as the sum of the identity matrix with the diagonal-times-Toeplitz matrices.2.An ADI-like iteration method is proposed for this type of linear system.The related theory states that the iteration method is convergent,and each iteration requires solving two linear subsystems with the Hessenberg coefficient matrix.This Both linear subsystems can be calculated by Hessenberg LU decomposition,and the ADIlike preconditioner is given.Numerical examples show that the iteration method and its preconditioner are both effective.3.The ADI-like iteration method is improved,and the Hessenberg matrix splitting iteration method is proposed.The theoretical analysis proves that the iteration method is unconditionally convergent.Numerical experiments also show that the iteration method and its preprocessor have better computing performance compared with the ADI-like iteration method and its preconditioner.