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

Abstract

In this paper, we consider the numerical method for solving the two-dimensional time-fractional convection-diffusion equation with a fractional derivative of order α\alpha (1<α<21\lt \alpha \lt 2). 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 H1{H}^{1} norm convergence of the scheme are proved rigorously. The convergence order is O(τ3−α+h14+h24)O\left({\tau }^{3-\alpha }+{h}_{1}^{4}+{h}_{2}^{4}), where τ\tau is the temporal grid size and h1{h}_{1}, h2{h}_{2} are spatial grid sizes in the xx and yy directions, respectively. It is proved that the method can even attain (1+α)\left(1+\alpha ) order accuracy in temporal for some special cases. Numerical results are presented to demonstrate the effectiveness of theoretical analysis.

Keywords

• 2d time-fractional convection-diffusion equation
• adi scheme
• unconditional stability
• convergence
• 37mxx
• 65-xx