A High Order Method on Graded Meshes for a Time-Fractional Diffusion Problem

摘要

In a recent paper we showed numerically and theoretically that a straightforward generalisation of Alikhanov's "L2-1_σ" scheme is O(M~(-2)) accurate on suitably chosen graded meshes (with M time intervals) for initial-value problems (IVPs) and initial-boundary value problems (IBVPs) with a Caputo fractional time derivative of order a, whose solutions typically exhibit a weak singularity at the initial time t = 0. The present paper constructs a better generalisation of Alikhanov's scheme that is demonstrated numerically to be O(M~(-(3-α))) accurate for these classes of IVPs and IBVPs, but its rigorous analysis remains an open problem.