Parallel two-grids algorithms for solution of anomalous diffusion equations of fractional order

New parallel algorithms are proposed for solving the initial-boundary value problems for anomalous diffusion equations with the Riemann-Liouville spatial- and/or time-fractional derivatives. A two-grid technique is employed to construct these algorithms. Spline-approximation on a coarse grid is used to compute the spatial and time long-range effects, and a fine grid is used for finite-difference discretization of the fractional diffusion equations. The parallel algorithms with a spatial and a time domain decomposition are discussed separately. The approach originally developed for the Parareal algorithm is used for time domain decomposition. The theoretical estimates of the speed-up and efficiency of the proposed algorithms are given. It has been shown that the algorithms have a superlinear speed-up in comparison with a classical sequential finite-difference algorithm, and have the same accuracy if the size of a fine grid is agreed with the size of a coarse grid. Some computational results are also presented to verify the efficiency of the proposed algorithms.