A fast second-order implicit difference method for time-space fractional advection-diffusion equation

In this paper, we consider a fast and second-order implicit difference method for approximation of a class of time-space fractional variable coefficients advection-diffusion equation. To begin with, we construct an implicit difference scheme, based on $L2-1_{\sigma}$ formula [A. A. Alikhanov, A new difference scheme for the time fractional diffusion equation, \emph{J. Comput. Phys.}, 280 (2015)] for the temporal discretization and weighted and shifted Gr\"{u}nwald method for the spatial discretization. Then, unconditional stability of the implicit difference scheme is proved, and we theoretically and numerically show that it converges in the $L_2$-norm with the optimal order $\mathcal{O}(\tau^2 + h^2)$ with time step $\tau$ and mesh size $h$. Secondly, three fast Krylov subspace solvers with suitable circulant preconditioners are designed to solve the discretized linear systems with the Toeplitz matrix. In each iterative step, these methods reduce the memory requirement of the resulting linear equations from $\mathcal{O}(N^2)$ to $\mathcal{O}(N)$ and the computational complexity from $\mathcal{O}(N^3)$ to $\mathcal{O}(N \log N)$, where $N$ is the number of grid nodes. Finally, numerical experiments are carried out to demonstrate that these methods are more practical than the traditional direct solvers of the implicit difference methods, in terms of memory requirement and computational cost.