A high-order difference scheme for the fractional sub-diffusion equation

ABSTRACT

Based on the Lubich's high-order operators, a second-order temporal finite-difference method is considered for the fractional sub-diffusion equation. It has been proved that the finite-difference scheme is unconditionally stable and convergent in $L^2$ norm by the energy method in both one- and two-dimensional cases. The rate of convergence is order of two in temporal direction under the initial value satisfying some suitable conditions. Some numerical examples are given to confirm the theoretical results.

KEYWORDS:

• fractional derivative
• multi-term
• Lubich's operator
• difference scheme
• compact
• convergence

2010 AMS SUBJECT CLASSIFICATIONS:

• 35R11
• 65M06
• 65M12
• 65M15

Acknowledgments

The authors are grateful to anonymous referees for their valuable comments and suggestions to improve this work.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The research is supported by National Natural Science Foundation of China [No. 11271068] and by the Fundamental Research Funds for the Central Universities and the Research and Innovation Project for College Graduates of Jiangsu Province [Grant No.: KYLX_0081]. G. Lin would like to thank the support of the Multifaceted Mathematics for Complex Energy Systems (M${}^2$ACS) project, the Collaboratory on Mathematics for Mesoscopic Modelling of Materials project and NSF Grant DMS-1115887.