Numerical Methods For The Two-dimensional Two-sided Space Fractional Advection-diffusion Equations

Advection-diffusion equation is a class of the basic eqauation of motion,it can be used to describe the air pollution,river pollution,nuclear waste pollution in the distribution of contaminants,the fluid flow and heat transfer fluids,and many other physical phenomena.However,compared with integer order differential equation,fractional differential equation is less perfect.It is short for scientific solution formula.At present the study of it is still in its infancy.Like integer order differential equation,only a few types of fractional differential equation can find out its analytic solution.In most cases,it can only use the numerical method to compute its solution.Therefore,the research of numerical evaluation has very important significance.In this paper,we mainly study the numerical methods for the two dimensional two-sided space fractional advection-diffusion equations.All the fractional derivatives refer to Riemann-Liouville definition in terms of the fractional derivative in this article.The main work is as follows: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.Some preliminary knowledge is given,including fractional derivative,Toeplitz and circulant matrices and the related theorem.In Chapter two,the finite difference method for the two dimensional two-sided s-pace fractional advection-diffusion equations is studied.According to the thought of the finite difference method proposed by some scholars,we construct the weighted Crank-Nicolson(CN)scheme for the equation which can achieve two order accuracy both in time and space.The method can guarantee that the coefficient matrix of the discrete sys-tem strictly diagonally dominant.And then we analyze the existence and uniqueness of the solution,stability and convergence of the format.Also the numerical example verifies the effectiveness,accuracy and reliability of the method.In Chapter three,Because of the nonlocal property of fractional differential operators,the numerical methods for advection-diffusion equations often generate dense or even full coefficient matrices.Consequently,the numerical solution of these methods often require computational work of O(N3)per time step and memory of O(N2)for where N is the number of grid points.We construct the alternating-direction implicit weighted Crank-Nicolson(ADI-CN)scheme for the equation,which can descend the dimension.The stability and convergence of the method are theoretically established.Finally,the numerical example verifies the effectiveness,accuracy and reliability of the method.In Chapter four,we study the fast iterative method for the two dimensional two-sided space fractional advection-diffusion equations.By the fast Fourier transform,the fast conjugate gradient squared method only requires storage of O(N)and computational work of O(N log N)per iteration,while retaining the same accuracy as the weighted CN scheme.Numerical experiment is presented to verify the efficiency and accuracy of the fast method.In Chapter five,we study the fast finite difference method for the two dimensional two-sided space fractional advection-diffusion equations.By the fast Fourier transform,the fast method only requires storage of O(N)and computational work of O(N log2 N)per time step,while retaining the same accuracy as the ADI-CN scheme.Numerical experiment is presented to verify the efficiency and accuracy of the fast method.