The accuracy and stability of CN-WSGD schemes for space fractional diffusion equation

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

卷:363

Article

Lin, Fu-Rong¹  Liu, Wei-Dong¹ 

[1] Shantou Univ, Dept Math, Shantou 515063, Guangdong, Peoples R China

关键词: Riemann-Liouville fractional derivative;    Fractional differential equation;    CN-WSGD scheme;    Accuracy;    Stability;   

DOI  :  10.1016/j.cam.2019.06.008

来源: Elsevier

【 摘 要 】

In this paper, we discuss the accuracy and the stability of the CN-WSGD when it is applied to the initial-boundary value problem for a space fractional diffusion equation. In the CN-WSGD scheme, the weighted and shifted Grunwald difference (WSGD) scheme is used to discretize the space fractional derivative and the Crank-Nicolson (CN) scheme is used to discretize the temporal derivative. There is a free parameter in a CN-WSGD scheme that affects the stability and the accuracy of the scheme. In general, a CN-WSGD scheme has temporally and spatially second order accuracy, and there exists a scheme that has spatially third order accuracy. However, the third order accuracy scheme is not always stable. In this paper, we present and prove some theoretical results (old and new) on the stability of several CN-WSGD schemes, and consider choosing a value for the parameter such that the corresponding CN-WSGD scheme is unconditioned stable and has optimal upper bound for the accuracy. Numerical results are presented to verify the theoretical results. (C) 2019 The Authors. Published by Elsevier B.V.

【 授权许可】

Free