A fully discrete GL-ADI scheme for 2D time-fractional reaction-subdiffusion equation

Alternating direction implicit (ADI) difference method for solving a 2D reaction-subdiffusion equation whose solution behaves a weak singularity at $t=0$ is studied in this paper. A Grünwald-Letnikov (GL) approximation is used for the discretization of Caputo fractional derivative (of order α, with $0<\alpha <1$) on a uniform mesh. Stability and convergence of the fully discrete ADI scheme are rigorously established. With the help of a discrete fractional Gronwall inequality, we get the sharp error estimate. The stability in $L^2$ norm and the convergence of the GL-ADI scheme are strictly proved, where the convergent order is $O(\tau t_s^{\alpha -1}+{\tau }^{2\alpha }+h_1^2+h_2^2)$. Numerical experiments are given to verify the theoretical analysis.