Numerical methods for time-fractional convection-diffusion problems with high-order accuracy

In this paper, we consider the numerical method for solving the two-dimensional time-fractional convection-diffusion equation with a fractional derivative of order ( ). By combining the compact difference approach for spatial discretization and the alternating direction implicit (ADI) method in the time stepping, a compact ADI scheme is proposed. The unconditional stability and norm convergence of the scheme are proved rigorously. The convergence order is , where is the temporal grid size and , are spatial grid sizes in the and directions, respectively. It is proved that the method can even attain order accuracy in temporal for some special cases. Numerical results are presented to demonstrate the effectiveness of theoretical analysis.