修正的晶体相场方程的无条件能量稳定数值格式

doi:10.3969/j.issn.1006-8074.2023.04.004

数学理论与应用 ›› 2023, Vol. 43 ›› Issue (4): 59-75.doi: 10.3969/j.issn.1006-8074.2023.04.004

修正的晶体相场方程的无条件能量稳定数值格式

梁译泓¹, 贾宏恩^1,2,*

1.太原理工大学数学学院, 晋中, 030600; 2.智能优化计算与区块链技术山西省重点实验室, 晋中, 030619

出版日期:2023-12-28 发布日期:2024-01-03
通讯作者: 贾宏恩(1981–), 教授, 博士, 从事偏微分方程理论及其数值计算; E-mail: jiahongen@aliyun.com
基金资助:
山西省归国留学基金项目(No. 2021-029)和山西省科技合作交流专项项目(No. 202104041101019)资助

An Unconditionally Energy Stable Numerical Scheme for the Modified Phase Field Crystal Equation

Liang Yihong¹, Jia Hongen^1,2,*

1. School of Mathematics, Taiyuan University of Technology, Jinzhong 030600, China; 2.Shanxi Key Laboratory for Intelligent Optimization Computing and Blockchain Technology, Jinzhong 030619, China

Online:2023-12-28 Published:2024-01-03

摘要/Abstract

摘要： 针对具有周期边界条件的修正的晶体相场方程，本文构建一个线性、二阶、无条件能量稳定的时间半离散数值格式, 通过引入拉格朗日乘子处理非线性项, 使用~Crank-Nicolson~方法进行时间离散, 依次证明该数值格式的唯一可解性、无条件能量稳定性及在时间上的二阶无条件收敛性, 最后通过数值算例对该格式的有效性进行验证.

关键词: 修正的晶体相场方程, 线性格式, 无条件能量稳定, 误差估计

Abstract:

This paper constructs a linear, second-order, unconditionally energy stable, semi-discrete time stepping scheme for the modified phase field crystal equation with periodic boundary conditions.

The unique solvability, unconditionally energy stability and unconditionally temporal convergence of order 2 of the numerical scheme are showed by introducing a Lagrange multiplier to deal with the nonlinear terms and adopting the second-order

Crank-Nicolson method to discrete time. Numerical experiments are given in the last section to validate the efficiency of the proposed scheme.

Key words: Modified phase field crystal equation, Linear scheme, Unconditionally energy stability, Error estimate

梁译泓, 贾宏恩. 修正的晶体相场方程的无条件能量稳定数值格式[J]. 数学理论与应用, 2023, 43(4): 59-75.

Liang Yihong, Jia Hongen. An Unconditionally Energy Stable Numerical Scheme for the Modified Phase Field Crystal Equation[J]. Mathematical Theory and Applications, 2023, 43(4): 59-75.

[1]	王为, 王晚生. 延迟抛物型方程二阶BDF方法的稳定性和后验误差估计[J]. 数学理论与应用, 2017, 37(3-4): 26-37.
[2]	赵新阳, 王晚生. 抛物型方程变步长BDF₂方法的后验误差估计[J]. 数学理论与应用, 2017, 37(1): 32-37.

修正的晶体相场方程的无条件能量稳定数值格式

An Unconditionally Energy Stable Numerical Scheme for the Modified Phase Field Crystal Equation

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