A Fast Second-Order Implicit Difference Method for Time-Space Fractional Advection-Diffusion Equation

In this paper, we consider a fast second-order implicit difference method to approximate a class of linear time-space fractional variable coefficients advection-diffusion equation. To begin with, an implicit difference scheme is constructed based on L2-1σ formula [Alikhanov AA. 2015;280:424–38.] for the temporal discretization and weighted and shifted Grünwald method for the spatial discretization. Then, the unconditional stability of the scheme is proved. We theoretically and numerically show that it converges in the L2-norm with the optimal order O(τ2+h2) with the time step τ and mesh size h. Moreover, the same technique is utilized to solve the nonlinear case of this problem. For the purpose of effectively solving these discretized systems, which have Toeplitz structure, two fast Krylov subspace solvers with suitable circulant preconditioners are designed. In each iterative step, these methods reduce the storage requirements of these discretized systems from O(N2) to O(N) and the computational complexity from O(N3) to O(N log N), where N is the number of grid nodes. Numerical experiments are carried out to demonstrate that these methods are more practical than the traditional direct solvers of the implicit difference methods, in aspects of memory requirement and calculation time.