Alternating direction implicit schemes for the two-dimensional fractional sub-diffusion equation

New numerical techniques are presented for the solution of a two-dimensional anomalous sub-diffusion equation with time fractional derivative. In these methods, standard central difference approximation is used for the spatial discretization, and, for the time stepping, two new alternating direction implicit (ADI) schemes based on the L₁ approximation and backward Euler method are considered. The two ADI schemes are constructed by adding two different small terms, which are different from standard ADI methods. The solvability, unconditional stability and H¹ norm convergence are proved. Numerical results are presented to support our theoretical analysis and indicate the efficiency of both methods.