  In this paper, a second order finite difference scheme is investigated for
time-dependent one-side space fractional diffusion equations with variable
coefficients. The existing schemes for the equation with variable coefficients
have temporal convergence rate no better than second order and spatial
convergence rate no better than first order, theoretically. In the presented
scheme, the Crank-Nicolson temporal discretization and a second-order
weighted-and-shifted Gr\"unwald-Letnikov spatial discretization are employed.
Theoretically, the unconditional stability and the second-order convergence in
time and space of the proposed scheme are established under some conditions on
the diffusion coefficients. Moreover, a Toeplitz preconditioner is proposed for
linear systems arising from the proposed scheme. The condition number of the
preconditioned matrix is proven to be bounded by a constant independent of the
discretization step-sizes so that the Krylov subspace solver for the
preconditioned linear systems converges linearly. Numerical results are
reported to show the convergence rate and the efficiency of the proposed
scheme.
