A fast second-order implicit scheme for non-linear time-space fractional diffusion equation with time delay and drift term

In this paper, a second-order accurate implicit scheme based on the L2-1σ formula for temporal variable and the fractional centered difference formula for spatial discretization is established to solve a class of time-space fractional diffusion equations with time drift term and non-linear delayed source function. The stability of this scheme is proved rigorously by the discrete energy method under several auxiliary assumptions, then we theoretically and numerically show that the proposed scheme converges in the L2-norm with the order O((Δt)2+h2) with time step Δt and mesh size h. Moreover, it finds that the discreted linear systems are symmetric Toeplitz systems. In order to solve these systems efficiently, the conjugate gradient method with suitable circulant preconditioners is designed. In each iterative step, the storage requirements and the computational complexity of the resulting equations are O(N) and O(NlogN) respectively, where N is the number of grid nodes. Numerical experiments are carried out to demonstrate the effectiveness of our proposed circulant preconditioners.