  The aim of this paper is to develop fast second-order accurate difference
schemes for solving one- and two-dimensional time distributed-order and Riesz
space fractional diffusion equations. We adopt the same measures for one- and
two-dimensional problems as follows: we first transform the time
distributed-order fractional diffusion problem into the multi-term time-space
fractional diffusion problem with the composite trapezoid formula. Then, we
propose a second-order accurate difference scheme based on the interpolation
approximation on a special point to solve the resultant problem. Meanwhile, the
unconditional stability and convergence of the new difference scheme in
$L_2$-norm are proved. Furthermore, we find that the discretizations lead to a
series of Toeplitz systems which can be efficiently solved by Krylov subspace
methods with suitable circulant preconditioners. Finally, numerical results are
presented to show the effectiveness of the proposed difference methods and
demonstrate the fast convergence of our preconditioned Krylov subspace methods.
