A High-Order Compact Finite Difference Scheme for the Fractional Sub-diffusion Equation

Based on the weighted and shifted Grunwald operator, a new high-order compact finite difference scheme is derived for the fractional sub-diffusion equation. It is proved that the difference scheme is unconditionally stable and convergent in L-infinity-norm by the energy method. The convergence order is O(tau(3) + h(4)), where tau is the temporal step size and h is the spatial step size. Although the unconditional stability and convergence of the difference scheme are obtained for all alpha is an element of (0, alpha*], where alpha* = 0.9569347, some numerical experiments show that they are valid for all alpha is an element of (0, 1). Finally, some numerical examples are given to confirm the theoretical results.