A fast Alikhanov algorithm with general nonuniform time steps for a two-dimensional distributed-order time-space fractional advection-dispersion equation

In this paper, we propose a fast Alikhanov algorithm with nonuniform time steps for a two dimensional distributed-order time-space fractional advection-dispersion equation. First, an efficient fast Alikhanov algorithm on the general nonuniform time steps for the evaluation of Caputo fractional derivative is presented to sharply reduce the computational work and storage, and are applied to the distributed-order time fractional derivative or multi-term time fractional derivative under the nonsmooth regularity assumptions. And a generalized discrete fractional Gronwall inequality is extended to multi-term fractional derivative or distributed-order fractional derivative for analyzing theoretically our algorithm. Then the stability and convergence of time semi-discrete scheme are investigated. Furthermore, we derive the corresponding fully discrete scheme by finite element method and discuss its convergence. At last, the given numerical examples adequately confirm the correctness of theoretical analysis and compare the computing effectiveness between the fast algorithm and the direct method.