[Paper Review] A Second-Order Crank-Nicolson Method for Subdiffusion

Bangti Jin, Buyang Li+1 more|arXiv (Cornell University)|Jul 23, 2016

Fractional Differential Equations Solutions20 references3 citations

TL;DR

This paper proposes a second-order accurate Crank-Nicolson-type time discretization for the subdiffusion equation with a Caputo fractional derivative of order $\alpha \in (0,1)$. By combining backward Euler convolution quadrature with a $\theta$-method where $\theta = \alpha/2$, and applying initial corrections, the scheme achieves second-order convergence in time for both smooth and nonsmooth data when coupled with the Galerkin finite element method in space.

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 combines the backward Euler convolution quadrature with a $ heta$-type method, with the parameter $ heta$ dependent on the fractional order $\alpha$ by $ heta=\alpha/2$, and it naturally generalizes the classical Crank-Nicolson method. We develop essential initial corrections for the fractional Crank-Nicolson scheme, and together with the Galerkin finite element method in space, obtain a fully discrete scheme. 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 further, the comparative study indicates that the scheme is competitive with existing schemes.

Motivation & Objective

• To develop a time discretization scheme with second-order accuracy for subdiffusion equations involving a Caputo fractional derivative of order $\alpha \in (0,1)$.
• To generalize the classical Crank-Nicolson method to fractional subdiffusion by introducing a $\theta$-dependent parameter $\theta = \alpha/2$.
• To address the loss of convergence order for nonsmooth initial data by introducing essential initial corrections.
• To combine the proposed time scheme with the Galerkin finite element method in space to form a fully discrete scheme.
• To provide a complete error analysis and demonstrate the scheme's robustness and efficiency through extensive numerical experiments.

Proposed method

• The time discretization uses a Crank-Nicolson-type scheme based on backward Euler convolution quadrature with a $\theta$-method where $\theta = \alpha/2$, ensuring generalization of the classical method to fractional order.
• Initial corrections are derived and applied to restore second-order convergence for nonsmooth problem data, which otherwise degrades the convergence rate.
• The spatial discretization employs the Galerkin finite element method, leading to a fully discrete scheme with optimal stability and convergence properties.
• The error analysis is conducted in both $L^2$ and energy norms, establishing second-order convergence in time for both smooth and nonsmooth initial data.
• The scheme is validated through extensive numerical experiments, including comparisons with existing methods, demonstrating its accuracy, efficiency, and robustness.

Experimental results

Research questions

• RQ1 Can a Crank-Nicolson-type scheme be constructed for subdiffusion with a Caputo fractional derivative that achieves second-order accuracy in time?
• RQ2 How can initial corrections be designed to restore second-order convergence for nonsmooth initial data in fractional subdiffusion schemes?
• RQ3 Does the proposed scheme maintain second-order convergence when combined with the Galerkin finite element method in space?
• RQ4 How does the proposed scheme compare in accuracy and efficiency to existing numerical schemes for subdiffusion?
• RQ5 Can the scheme be generalized from classical Crank-Nicolson by adjusting the $\theta$ parameter based on the fractional order $\alpha$?

Key findings

• The proposed scheme achieves second-order convergence in time for both smooth and nonsmooth initial data, overcoming the typical order reduction in fractional subdiffusion.
• The initial corrections are essential to maintain second-order accuracy, especially for nonsmooth data, and are rigorously derived in the analysis.
• The fully discrete scheme, combining the time scheme with the Galerkin finite element method, is unconditionally stable and convergent with optimal order.
• Numerical experiments confirm the theoretical error estimates and demonstrate the scheme’s robustness across various test cases.
• The comparative study shows the scheme is competitive with existing methods in terms of accuracy and computational efficiency.
• The parameter choice $\theta = \alpha/2$ effectively generalizes the classical Crank-Nicolson method to the fractional subdiffusion setting.