A variable-step high-order scheme for time-fractional advection-diffusion equation with mixed derivatives

We consider a high accuracy numerical scheme for solving the two-dimensional time-fractional advection-diffusion equation including mixed derivatives, where the variable-step Alikhanov formula and a fourth-order compact approximation are employed to time and space derivatives, respectively. Under mild assumptions on the time step-sizes, we obtain the unconditional stability and high-order convergence (second-order in time and fourth-order in space) of the proposed scheme by energy method. The theoretical statements are justified by the numerical experiments.