Some Numerical Methods For The Two-sided Space Fractional Advection-diffusion Equations With Constant Coefficients

Fractional differential equations are equations with fractional differentials,and the memory and genetic properties of fractional differential operators make it possible to sim-ulate complex processes such as certain natural physical phenomena and motion changes of matter better.Therefore,fractional differential equations are favored by scholars in different scientific fields such as fluid mechanics,physics,chemistry,and biology.How-ever,the current research on fractional differential equations is still in its infancy,and only a few fractional differential equations can obtain analytical solutions.Therefore,the research on numerical solution of fractional differential equations is of great significance.In this paper,the Riemann-Liouville type one-dimensional two-sided space fractional convection-diffusion equation with constant coefficient is studied,and the numerical solu-tion of the equation is taken as the research goal.The main work is as follow:In Chapter one,an introduction to the history of the fractional calculus,the research significance of fractional differential equation and the current research status at home and abroad of the numerical method for fractional differential equations are given.In Chapter two,some preliminary knowledge is given,including the definition of frac-tional derivatives,Toeplitz matrices and circulant matrices,and some common function spaces and operators.In the third chapter,combined with the research results of the one-dimensional two-sided space fractional convection-diffusion equation with constant coefficient in recent years,the finite difference method is used to construct the second-order precision center weighted scheme in both time and space,the stability and convergence of the format are analyzed.Finally,a numerical example is given to verify the validity and accuracy of the scheme and compare it with other second-order discrete scheme.However,the central weighted scheme cannot guarantee that the coefficient matrix is strictly diagonally dominant.When the split is very fine or the spatial domain is large,it brings some difficulties to the solution.In the fourth chapter,in order to improve the defects of the central weighted scheme,a new weighted scheme is proposed.This scheme achieves second-order accuracy when the spatial step size is equal to the time step,and can guarantee that the coefficient matrix is strictly diagonally dominant.Then the existence and uniqueness of the solution,stability and convergence of the scheme are analyzed.Finally,numerical examples are given to verify the validity and accuracy of the format.In the fifth chapter,after fully studying the structure of coefficient matrix of central weighted scheme and the new weighted scheme,a fast finite difference method for the one-dimensional two-sided space fractional convection-diffusion equation with constant coefficients is developed.The fast algorithm maintains the same accuracy as the original format.Under the premise,the fast Fourier transform requires only O(M)storage and O(M log~2M)calculation for each time step.Finally,numerical examples are given to verify the effectiveness and accuracy of the fast algorithm.