A Corrected L1 Method for a Time-Fractional Subdiffusion Equation

Piecewise linear interpolation has been widely applied to discretize the Caputo fractional derivative operator that yields the widely used L1 method. In this paper, a corrected L1 method is proposed for discretizing the Caputo fractional derivative operator, which achieves the similar level of accuracy as that of the L1 method. The corrected L1 method is applied to solve the fractional initial value and time-fractional subdiffusion problems, the convergence of the corresponding numerical schemes is rigorous proved. In order to reduce the storage and computational cost caused by the nonlocality of the Caputo fractional operator, a fast memory-saving corrected L1 method is developed. It is proved that the difference between the solution of the corrected L1 method and the fast corrected L1 method can be made arbitrarily small and is independently the sizes of the time and/or space grids.Numerical results support the theoretical analysis.