A high-order difference scheme for the fractional sub-diffusion equation

Based on the Lubich's high-order operators, a second-order temporal finite-difference method is considered for the fractional sub-diffusion equation. It has been proved that the finite-difference scheme is unconditionally stable and convergent in L-2 norm by the energy method in both one- and two-dimensional cases. The rate of convergence is order of two in temporal direction under the initial value satisfying some suitable conditions. Some numerical examples are given to confirm the theoretical results.