Electron. J. Differential Equations, Vol. 2019 (2019), No. 76, pp. 1-17.

Crank-Nicolson Legendre spectral approximation for space-fractional Allen-Cahn equation

Wenping Chen, Shujuan Lu, Hu Chen, Haiyu Liu

Abstract:
In this article, we consider spectral methods for solving the initial-boundary value problem of the space fractional-order Allen-Cahn equation. A fully discrete scheme based on the modified Crank-Nicolson scheme in time and the Legendre spectral method in space is established. The existence and uniqueness of the fully discrete scheme are derived, and the stability and convergence analysis of the fully discrete scheme are proved rigorously. By constructing a fractional duality argument, the corresponding optimal error estimates in $L^2$ and $H^\alpha$ norm are derived, respectively. Also, numerical experiments are performed to support the theoretical results.

Submitted August 28, 2018. Published May 31, 2019.
Math Subject Classifications: 35R11, 65M06, 65M70, 65M12.
Key Words: Space-fractional Allen-Cahn equation; Legendre spectral method; modified Crank-Nicolson scheme; stability; convergence.

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Wenping Chen
School of Mathematics and Systems Science & LMIB
Beijing University of Aeronautics & Astronautics
Beijing, 100191, China
email: anhuicwp@163.com
  Shujuan Lu
School of Mathematics and Systems Science & LMIB
Beijing University of Aeronautics & Astronautics
Beijing, 100191, China
email: lsj@buaa.edu.cn
Hu Chen
Beijing Computational Science Research Center
Beijing, 100193, China
email: chenhuwenlong@126.com
Haiyu Liu
School of Mathematics and Systems Science & LMIB
Beijing University of Aeronautics & Astronautics
Beijing, 100191, China
email: liuhaiyu0415@163.com

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