High-Order Compact Finite Difference Methods For Fractional Sub-diffusion Equations With Variable Coefficients

Fractional partial differential equations are widely used in various fields of science and engineering.In this thesis,we establish several high-order compact finite difference methods for the initial-boundary value problems of fractional sub-diffusion equations with variable coefficients,and give the corresponding nu-merical analysis.This thesis consists of six chapters as follows.In Chapter 1,we briefly introduce some related research background and the motivations of the thesis.In Chapter 2,we construct and analyze compact finite difference methods for a class of time-fractional convection-reaction-diffusion equations with spatial-ly variable convection and reaction coefficients.Based on some new techniques coupled with the L2-1? approximation formula of the time-fractional derivative and a fourth-order compact finite difference approximation to the spatial deriva-tive,a compact finite difference method is proposed for the equations.The local truncation error and the solvability of the method are discussed in detail.The unconditional stability of the method and also its convergence of second-order in time and fourth-order in space are rigorously proved using a discrete energy analysis method.The proposed method is further extended to the more general case when the convection and reaction coefficients are variable both spatially and temporally.A combined compact finite difference method with high-order accu-racy is also proposed.Numerical results demonstrate the effectiveness of these methods.In Chapter 3,we continue studying the variable coefficient fractional sub-diffusion equation considered in Chapter 2.Based on the weighted and shifted Grunwald-Letnikov formula for the time-fractional derivative and a compact fi-nite difference approximation for the spatial derivative,we establish an uncon-ditionally stable compact difference method.The local truncation error and the solvability of the resulting scheme are discussed in detail.The stability of the method and its convergence of third-order in time and fourth-order in space are rigorously proved by the discrete energy method.Combining this method with a Richardson extrapolation,we present an extrapolated compact difference method which is fourth-order accurate in both time and space.A rigorous proof for the convergence of the extrapolation method is given.Numerical results confirm our theoretical analysis,and demonstrate the accuracy of the compact difference method and the effectiveness of the extrapolated compact difference method.In Chapter 4,we propose and analyze a high-order compact finite difference method for solving a class of variable coefficient time-fractional sub-diffusion e-quations in conservative form.The ?th-order Caputo time-fractional deriva-tive(? ?(0,1))is discretized by L2 formula which is constructed by piecewise quadratic interpolating polynomials.The variable coefficient spatial differential operator is approximated by a fourth-order compact finite difference operator.By developing a technique of discrete energy analysis,a full theoretical analysis of the stability and convergence of the method is carried out for the general case of variable coefficient and for all ? ?(0,1).The optimal error estimate is obtained in the L2 norm and shows that the proposed method has the temporal(3-?)th-order accuracy and the spatial fourth-order accuracy.Further approximations are also considered for enlarging the applicability of the method.Applications are given to three model problems,and numerical results are presented to demon-strate the theoretical analysis results.In Chapter 5,we continue studying the variable coefficient fractional sub-diffusion equation considered in Chapter 4.Based on the Lubich difference op-erator for the time-fractional derivative and a compact finite difference approxi-mation for the spatial derivative,we establish a set of compact finite difference methods.The proposed methods have the global convergence order O(?r + h4),where r>2 is a positive integer and ? and h are the temporal and spatial step-s.Such new high-order compact difference methods greatly improve the known methods in the literature.The local truncation error and the solvability of the methods are discussed in detail.By applying a discrete energy technique to the matrix form of the methods,a rigorous theoretical analysis of the stability and convergence of the methods is carried out and the optimal error estimates in the weighted ?1,L2 and L? norms are obtained for 2 ? r ? 6 and for the general case of variable coefficient.Applications are given to two model problems,and some numerical results are presented to illustrate the various convergence orders of the methods.In Chapter 6,we summarise the main results of the thesis and outline the works which are deserved to be studied in the future.