In this paper we want to propose practical numerical methods to solve a class of initial-boundary problem of time-space fractional convection-diffusion equations (TSFCDEs). To start with, an implicit difference method based on two-sided weighted shifted Grünwald formulae is proposed with a discussion of the stability and convergence. We construct an implicit difference scheme (IDS) and show that it converges with second order accuracy in both time and space. Then, we develop fast solution methods for handling the resulting system of linear equation with the Toeplitz matrix. The fast Krylov subspace solvers with suitable circulant preconditioners are designed to deal with the resulting Toeplitz linear systems. * Each time level of these methods reduces the memory requirement of the proposed implicit difference scheme from O(N 2 ) to O(N ) and the computational complexity from O(N 3 ) to O(N log N ) in each iterative step, where N is the number of grid nodes. Extensive numerical example runs show the utility of these methods over the traditional direct solvers of the implicit difference methods, in terms of computational cost and memory requirements.

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It is known that the restarted full orthogonalization method (FOM) outperforms the restarted generalized minimum residual (GMRES) method in several circumstances for solving shifted linear systems when the shifts are handled simultaneously. Many variants of them have been proposed to enhance their performance. We show that another restarted method , the restarted Hessenberg method [M. Heyouni, Méthode de Hessenberg Généralisée et Applications (Ph.D. Thesis), Université des Sciences et Technologies de Lille, France, 1996] based on Hessenberg procedure, can effectively be employed, which can provide accelerating convergence rate with respect to the number of restarts. Theoretical analysis shows that the new residual of shifted restarted Hessenberg method is still collinear with each other. In these cases where the proposed algorithm needs less enough elapsed CPU time to converge than the earlier established restarted shifted FOM, the weighted restarted shifted FOM, and some other popular shifted iterative solvers based on the short-term vector recurrence, as shown via extensive numerical experiments involving the recently popular application of handling time fractional differential equations.