Novel energy dissipative method on the adaptive spatial discretization for the Allen-Cahn equation

doi:10.1088/1674-1056/abe37b

Abstract We propose a novel energy dissipative method for the Allen-Cahn equation on nonuniform grids. For spatial discretization, the classical central difference method is utilized, while the average vector field method is applied for time discretization. Compared with the average vector field method on the uniform mesh, the proposed method can involve fewer grid points and achieve better numerical performance over long time simulation. This is due to the moving mesh method, which can concentrate the grid points more densely where the solution changes drastically. Numerical experiments are provided to illustrate the advantages of the proposed concrete adaptive energy dissipative scheme under large time and space steps over a long time.

Keywords: moving mesh energy dissipative average vector field method Allen-Cahn equation

Received: 25 December 2020
Revised: 23 January 2021
Accepted manuscript online: 05 February 2021

PACS:	02.60.Cb	(Numerical simulation; solution of equations)
	05.70.Fh	(Phase transitions: general studies)
	02.70.Bf	(Finite-difference methods)

Fund: Project supported by the National Key R&D Program of China (Grant No. 2020YFA0709800), the National Natural Science Foundation of China (Grant Nos. 11901577, 11971481, 12071481, and 12001539), the Natural Science Foundation of Hunan, China (Grant Nos. S2017JJQNJJ0764 and 2020JJ5652), the fund from Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering (Grant No. 2018MMAEZD004), the Basic Research Foundation of National Numerical Wind Tunnel Project, China (Grant No. NNW2018-ZT4A08), and the Research Fund of National University of Defense Technology (Grant No. ZK19-37).

Corresponding Authors: Xu Qian
E-mail: qianxu@nudt.edu.cn

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Jing-Wei Sun(孙竟巍), Xu Qian(钱旭), Hong Zhang(张弘), and Song-He Song(宋松和) Novel energy dissipative method on the adaptive spatial discretization for the Allen-Cahn equation 2021 Chin. Phys. B 30 070201

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[1]	Average vector field methods for the coupled Schrödinger–KdV equations Zhang Hong (张弘), Song Song-He (宋松和), Chen Xu-Dong (陈绪栋), Zhou Wei-En (周炜恩). Chin. Phys. B, 2014, 23(7): 070208.
[2]	A high order energy preserving scheme for the strongly coupled nonlinear Schrödinger system Jiang Chao-Long (蒋朝龙), Sun Jian-Qiang (孙建强). Chin. Phys. B, 2014, 23(5): 050202.

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Novel energy dissipative method on the adaptive spatial discretization for the Allen-Cahn equation

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