An analysis of the crank-nicolson method for subdiffusion

In this work, we analyse a Crank-Nicolson type time-stepping scheme for the subdiffusion equation, which involves a Caputo fractional derivative of order α ∈ (0, 1) in time. It hybridizes the backward Euler convolution quadrature with a θ-type method, with the parameter θ dependent on the fractional order α by θ = α/2 and naturally generalizes the classical Crank-Nicolson method. We develop essential initial corrections at the starting two steps for the Crank-Nicolson scheme and, together with the Galerkin finite element method in space, obtain a fully discrete scheme. The overall scheme is easy to implement and robust with respect to data regularity. A complete error analysis of the fully discrete scheme is provided, and a second-order accuracy in time is established for both smooth and nonsmooth problem data. Extensive numerical experiments are provided to illustrate its accuracy, efficiency and robustness, and a comparative study also indicates its competitive with existing schemes.