Second-order error analysis of the averaged L1 scheme L1 for time-fractional initial-value and subdiffusion problems

Fractional initial-value problems (IVPs) and time-fractional initial-boundary value problems (IBVPs), each with a Caputo temporal derivative of order α ∈ (0, 1), are considered. An averaged variant of the well-known L1 scheme is proved to be O ( N −2 ) convergent for IVPs on suitably graded meshes with N points, thereby improving the O ( N −(2− α ) ) convergence rate of the standard L1 scheme. The analysis relies on a delicate decomposition of the temporal truncation error that yields a sharp dependence of the order of convergence on the degree of mesh grading used. This averaged L1 scheme can be combined with a finite difference or piecewise linear finite element discretization in space for IBVPs, and under a restriction on the temporal mesh width, one gets again O ( N −2 ) convergence in time, together with O ( h 2 ) convergence in space where h is the spatial mesh width. Numerical experiments support our results.