A New Characteristic Finite Diffefence For Fractional Advection-Diffusion Equations And Fast Algorithms For Fractional Diffusion Problems

Fractional calculus is widely applied in mathematical and physical fields. Therefore, the numerical solution of fractional partial differential equations has been the focus of concern. This paper focuses on some space-fractional diffu-sion equations and their numerical methods. It is divided into four chapters.In Chapter1, we first consider the second-order convection-diffusion equa-tion. A new characteristic finite difference method using linear interpolation and the corresponding infinity-norm error estimate are established. Error anal-ysis and numerical example proved this method is of second order, with the convergence rate of O(Δt2+h2). Then we develop the new characteristic finite difference method for space-fractional convection-diffusion equations, by com-bining the spatial extrapolation technique, in which the new character method is used to improve the converge order of time, and the spatial extrapolation to improve the converge order of space. Finally we give a specific numerical example,which proved the second-order convergence rate of our method for space-fractional convection-diffusion equation.In Chapter2, we develop the finite volume method for the following un-steady space-fractional diffusion equation: We get the finite volume scheme by Crank-Nicolson method in order to keep the advantage of finite volume method. According to the features of the coef-ficient matrix, it can be effectively stored with an O(N) memory requirement. Furthermore, the coefficient matrix can be splitted into a narrow ribbon matrix on the left and a Toeplitz-like remainder on the right. Thus, the fast Fourier transform method can be used to get the product of cycle matrix and vector on the right, resulting in an O(Nlog4N) fast algorithm. We carry out numerical experiment to compare the performance of the newly developed method with the traditional calculation method. It can be shown from the comparison that our method has not only a smaller error, but also an improved calculation efficiency.In Chapter3, we give the finite difference scheme for the following steady state space fractional diffusion equation: Similar as in Chapter2, we develop an effective O(N) storage for the coeffi-cient matrix. A new iterative method called logN diagonal iterative method is established based on the idea of division of matrix and the fast calcula-tion method given in Chapter2. Then we carry out numerical experiment to compare the performance of the newly developed iteration method with Ja-cobi iteration method. It is clearly that the logN diagonal iteration is greatly improved in calculation efficiency.In Chapter4, we give the conclusions of this paper.