An Analysis of the Crank-Nicolson Method for Subdiffusion

TL;DR

This paper analyzes a Crank-Nicolson type scheme for subdiffusion equations involving fractional derivatives, establishing second-order accuracy and demonstrating robustness and efficiency through numerical experiments.

Contribution

It introduces a generalized Crank-Nicolson scheme with initial corrections for subdiffusion, providing a complete error analysis and confirming second-order temporal accuracy.

Findings

Second-order accuracy in time for smooth and nonsmooth data

Robustness of the scheme with respect to data regularity

Numerical experiments confirm efficiency and competitiveness

Abstract

In this work, we analyze 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.