Stability and convergence of finite difference schemes for a class of time-fractional sub-diffusion equations based on certain superconvergence ?
The superconvergent points of the traditional first-order GrnwaldLetnikov formula to approximate the RiemannLiouville fractional derivatives are used to construct the several finite difference schemes for solving a class of time-fractional sub-diffusion equations. The obtained schemes can achieve the global second-order accuracy in time, which is independent of the anomalous diffusion exponent ? (0<?<1) ( 0 < ? < 1 ) in the governing equation. The spatial second-order scheme and the spatial fourth-order compact scheme are presented, respectively. The strict stability and convergence analysis on the resulting difference schemes are given by the discrete energy method. The extension to the two-dimensional case is also discussed. The compact difference scheme with the convergence of second-order in time and fourth-order in space is derived along with the strict stability and convergence analysis. Several numerical experiments are implemented to illustrate the computational efficiency, numerical accuracy and robustness of the proposed schemes.
stability convergence finite difference scheme time-fractional sub-diffusion equations grnwaldletnikov formula