Fast Matrix Splitting Iteration Method And Band Preconditioner For The Linear System From Spatial Fractional Diffusion Equations

Due to the unique non-local properties of the fractional derivative,Fractional differential equations can better model the phenomena and processes with historical memory and spatial correlation in the field of science and engineering than integerorder differential equations.Therefore Fractional differential equations have been widely used in physics,chemistry,finance,image processing and other fields.In recent years,fractional differential equations have received extensive attention from numerous scholars.In particular,the spatial fractional diffusion equation can describe the abnormal diffusion process more accurately,but it is difficult to obtain the analytical solution of the complex function.Therefore,it is very necessary to study how to construct an efficient numerical solution.This paper mainly considers the one-dimensional spaced fractional diffusion equation.By using the shifted-Grünwald finite difference scheme to discrete the original problem,we obtained the linear system.Then approximate solution of the original problem is obtained by solving the discrete linear system.Because computational complexity and time-consuming by solving directly large and complex linear systems solution is large,the iterative method is usually selected.In the paper,based on the special structure of the coefficient matrix(diagonal matrix plus Toeplitz matrix),we construct an iterative method based on matrix splitting and propose a kind of preconditioner for solving the discrete linear system of the spatial fractional diffusion equation.At last,we give theoretical analysis and numerical results in the paper.The main work of this paper is as follows:1.The coefficient matrix of the discrete linear system is the diagonal matrix plus the Toeplitz matrix.Based on the circulant and skew-circulant splitting(CSCS split)of the Toeplitz matrix,by splitting diagonal matrix and introducing adjusting parameters,using the alternating two-step direction implicit iterative format and introducing two parameters,we construct an efficient iterative method—DSCS iterative method.2.We analyzed the convergence of the DSCS iterative method and discussed two parameters introduced in the DSCS method.Therefore,we obtained the optimal estimation form of the parameters.3.The main information of the coefficient matrix is contained in several diagonals near the main diagonal,and the matrix elements far away from the main diagonal tend to zero.The Toeplitz matrix in the coefficient matrix have attenuation properties.We propose a band-shaped preprocessing matrix to improve the calculation efficiency of GMRES method.4.It is proved that the DSCS iterative method and the Band-GMRES method are high efficiency by two numerical cases.Besides,it is also certified that the optimal estimation form of the parameters in the DSCS method is feasible and effective.