Compact ADI And LOD Methods For Time Fractional Partial Differential Equations

In recent years, numerical methods have become more and more importan-t in obtaining approximate solutions of fractional partial differential equations (FPDEs), because of their various application in science and engineering. This paper is devoted to developing the effective numerical methods for several class-es of time FPDEs, including fractional convection-subdiffusion equations with Dirichlet boundary conditions, fractional sub-diffusion equations with Neumann boundary conditions, modified anomalous fractional sub-diffusion equations and fractional diffusion-wave equations.Firstly, this work is concerned with numerical methods for a class of two-dimensional fractional convection-subdiffusion equations with a time Caputo frac-tional derivative of order a (0< a< 1). We first transform the original equation into a special and equivalent form, which is then discretized by a fourth-order compact finite difference approximation in the spatial directions and by an al-ternating direction implicit (ADI) approximation in the temporal direction. The resulting compact ADI scheme is uniquely solvable and unconditionally stable. The optimal error estimates in the weighted L∞, H1 and L2 norms are obtained, and show that the compact ADI method has the temporal accuracy of order min{1+α,2-α} and the fourth-order spatial accuracy. Applications using two model problems give numerical results that demonstrate the accuracy and the effectiveness of this new method.Secondly, a compact alternating direction implicit (ADI) finite difference method is proposed for a class of two-dimensional time fractional sub-diffusion equations with nonhomogeneous Neumann boundary conditions. The time frac-tional derivative is given in the Caputo sense with the order a (0< α< 1). The unconditional stability and convergence of the resulting scheme are rigorous-ly proved. The error estimates in the weighted L2- and L∞-norms are obtained and show that the proposed compact ADI method has the fourth-order spatial accuracy and the temporal accuracy of order min{2 - α,1+α}. Two Richard- son extrapolation algorithms are presented, respectively, for α ∈ (0,1/2) and α ∈ (1/2,1) to enhance the global temporal accuracy of the computed solution to the order max{2 - α,1+a}. A rigorous convergence analysis of the extrapola-tion algorithm for α ∈ (0.1/2) is given. Numerical results confirm our theoretical analysis, and demonstrate the accuracy of the compact ADI method and the high efficiency of the extrapolation algorithms.Thirdly, a Crank-Nicolson-type compact locally one-dimensional (LOD) fi-nite difference method is proposed for a class of two-dimensional modified anoma-lous sub-diffusion equations with two time Riemann-Liouville fractional deriva-tives of orders (1-α) and (1 -β) (0< α,β< 1). The resulting scheme consists of simple tridiagonal systems and all computations are carried out completely in one spatial direction as for one-dimensional problems. This property evidently enhances the simplicity of programming and reduces the computational complex-ities and the computational costs. The unconditional stability and convergence of the scheme are rigorously proved. The error estimates in the standard H1- and L2-norms and the weighted L∞-norm are obtained and show that the proposed compact LOD method has the accuracy of the order 2 min{α,β} in time and 4 in space. A Richardson extrapolation algorithm is presented to increase the tempo-ral accuracy to the order min{α+β,4min{α,β}} if α≠β and min{1+α,4α} if α= β. Numerical results confirm our theoretical analysis, and demonstrate the accuracy of the compact LOD method and the high efficiency of the extrapolation algorithm.Finally, this work is concerned with numerical methods for a class of multi-dimensional fractional diffusion-wave equations with a time fractional derivative of order α (1< α< 2). A compact locally one-dimensional (LOD) finite dif-ference method is proposed for the equations. The resulting scheme consists of one-dimensional tridiagonal systems, and all computations are carried out com-pletely in one spatial direction as for one-dimensional problems. The uncondi-tional stability and H1 norm convergence of the scheme are rigorously proved for the three-dimensional case. The error estimates show that the proposed compact LOD method converges with the order (3 - α) in time and 4 in space. Numerical results confirm our theoretical analysis and illustrate the effectiveness of this new method.