A compact ADI method and its extrapolation for time fractional sub-diffusion equations with nonhomogeneous Neumann boundary conditions

A compact alternating direction implicit (ADI) finite difference method is proposed for two-dimensional time fractional sub-diffusion equations with nonhomogeneous Neumann boundary conditions. The unconditional stability and convergence of the method is proved. The error estimates in the weighted $L^2$- and $L^{\infty }$-norms are obtained. The proposed method has the fourth-order spatial accuracy and the temporal accuracy of order $min\{2?\alpha ,1+\alpha \}$, where $\alpha \in (0,1)$ is the order of the fractional derivative. In order to further improve the temporal accuracy, two Richardson extrapolation algorithms are presented. Numerical results demonstrate the accuracy of the compact ADI method and the high efficiency of the extrapolation algorithms.