An analysis of the Crank–Nicolson method for subdiffusion

Abstract

In this work, we analyse a Crank-Nicolson type time-stepping scheme for the subdiffusion equation, which involves a Caputo fractional derivative of order |$\alpha\in (0,1)$| in time. It hybridizes the backward Euler convolution quadrature with a |$\theta$|-type method, with the parameter |$\theta$| dependent on the fractional order |$\alpha$| by |$\theta=\alpha/2$| 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.