A second order difference method combined with time two-grid algorithm for two-dimensional time-fractional Fisher equation

In this paper, a finite difference (FD) scheme based on time two-grid algorithm is proposed for solving the two-dimensional time-fractional Fisher equation (2D-TFFE). Firstly, the Caputo fractional derivative and the spatial derivative are discretized by the $ L2-1_\sigma $ L2-1 sigma formula and the central difference formula, respectively. In order to improve the efficiency of computation, the time two-grid algorithm is then constructed. Secondly, based on the cut-off function technic, stability and convergence of the time two-grid FD scheme are obtained by the energy method, and the global convergence order is $ O(\tau _{F}<^>2 +\tau _{C}<^>4 +h_x<^>2 +h_y<^>2) $ O(tau F2+tau C4+hx2+hy2), where $ \tau _{F} $ tau F and $ \tau _{C} $ tau C represent time-step sizes on the fine and coarse grid, respectively, while $ h_x $ hx and $ h_y $ hy represent the space-step sizes. Finally, numerical experiments are presented to show the feasibility and efficiency of the algorithm.