Local Discontinuous Galerkin Scheme for Space Fractional Allen-Cahn Equation

doi:10.1007/s42967-019-00034-9

Communications on Applied Mathematics and Computation ›› 2020, Vol. 2 ›› Issue (1): 73-91.doi: 10.1007/s42967-019-00034-9

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Local Discontinuous Galerkin Scheme for Space Fractional Allen-Cahn Equation

Can Li, Shuming Liu

College of Sciences, Xi'an University of Technology, Xi'an 710048, China

收稿日期:2019-01-06 修回日期:2019-05-07 出版日期:2020-03-20 发布日期:2020-02-19
通讯作者: Can Li, Shuming Liu E-mail:mathlican@xaut.edu.cn;xaut_lsm@163.com
基金资助:
The authors would like to thank the referees for their valuable comments and suggestions that have vastly improved the original manuscript of this paper. The research is supported by the National Natural Science Foundations of China (Grant number 11426174) and the Natural Science Basic Research Plan in Shaanxi Province of China (Grant number 2018JM1016).

Local Discontinuous Galerkin Scheme for Space Fractional Allen-Cahn Equation

Can Li, Shuming Liu

College of Sciences, Xi'an University of Technology, Xi'an 710048, China

Received:2019-01-06 Revised:2019-05-07 Online:2020-03-20 Published:2020-02-19
Contact: Can Li, Shuming Liu E-mail:mathlican@xaut.edu.cn;xaut_lsm@163.com
Supported by:
The authors would like to thank the referees for their valuable comments and suggestions that have vastly improved the original manuscript of this paper. The research is supported by the National Natural Science Foundations of China (Grant number 11426174) and the Natural Science Basic Research Plan in Shaanxi Province of China (Grant number 2018JM1016).

摘要/Abstract

摘要： This paper is concerned with the efcient numerical solution for a space fractional Allen-Cahn (AC) equation. Based on the features of the fractional derivative, we design and analyze a semi-discrete local discontinuous Galerkin (LDG) scheme for the initial-boundary problem of the space fractional AC equation. We prove the optimal convergence rates of the semi-discrete LDG approximation for smooth solutions. Finally, we test the accuracy and efciency of the designed numerical scheme on a uniform grid by three examples. Numerical simulations show that the space fractional AC equation displays abundant dynamical behaviors.

关键词: Fractional Allen-Cahn equation, Local discontinuous Galerkin scheme, Error estimates

Abstract: This paper is concerned with the efcient numerical solution for a space fractional Allen-Cahn (AC) equation. Based on the features of the fractional derivative, we design and analyze a semi-discrete local discontinuous Galerkin (LDG) scheme for the initial-boundary problem of the space fractional AC equation. We prove the optimal convergence rates of the semi-discrete LDG approximation for smooth solutions. Finally, we test the accuracy and efciency of the designed numerical scheme on a uniform grid by three examples. Numerical simulations show that the space fractional AC equation displays abundant dynamical behaviors.

Key words: Fractional Allen-Cahn equation, Local discontinuous Galerkin scheme, Error estimates

中图分类号:

Can Li, Shuming Liu. Local Discontinuous Galerkin Scheme for Space Fractional Allen-Cahn Equation[J]. Communications on Applied Mathematics and Computation, 2020, 2(1): 73-91.

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Local Discontinuous Galerkin Scheme for Space Fractional Allen-Cahn Equation

Local Discontinuous Galerkin Scheme for Space Fractional Allen-Cahn Equation

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