Fast second-order accurate difference schemes for time distributed-order and Riesz space fractional diffusion equations

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.