Error analysis of a high-order compact ADI method for two-dimensional fractional convection-subdiffusion equations

This paper is concerned with numerical methods for a class of two-dimensional fractional convection-subdiffusion equations with a time Caputo fractional derivative of order $$\alpha (0<\alpha <1)$$α(0<α<1). We first transform the original equation into a special and equivalent form, which is then discretized by a fourth-order compact finite difference approximation in the spatial directions and by an alternating direction implicit (ADI) approximation in the temporal direction. The resulting compact ADI scheme is uniquely solvable and unconditionally stable. The optimal error estimates in the weighted $$L^{\infty }$$L∞, $$H^{1}$$H1 and $$L^{2}$$L2 norms are obtained, and show that the compact ADI method has the temporal accuracy of order $$\min \{1+\alpha ,2-\alpha \}$$min{1+α,2-α} and the fourth-order spatial accuracy. Applications using three model problems give numerical results that demonstrate the accuracy and the effectiveness of this new method.