Numerical approximations of Allen-Cahn and Cahn-Hilliard equations
Numerical approximations of Allen-Cahn and Cahn-Hilliard equations
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DOI：
10.3934/dcds.2010.28.1669
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发表时间：
2010-06
影响因子：
1.1
通讯作者：
Jie Shen;Xiaofeng Yang
中科院分区：
文献类型：
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作者：
Jie Shen;Xiaofeng Yang
对Allen-Cahn方程和Cahn-Hilliard方程的几种常用数值格式进行了稳定性分析和误差估计。结果表明，在半离散格式中，我们所考虑的所有格式要么是无条件能量稳定的，要么是具有合理稳定性条件的条件能量稳定的格式。文中还给出了所选格式在谱Galerkin近似下的误差估计。稳定性分析和误差估计都是基于弱列式的，因此结果可以很容易地推广到其他空间离散，如基于弱列式的Galerkin有限元方法。
Stability analyses and error estimates are carried out for a number of commonly used numerical schemes for the Allen-Cahn and Cahn-Hilliard equations. It is shown that all the schemes we considered are either unconditionally energy stable, or conditionally energy stable with reasonable stability conditions in the semi-discretized versions. Error estimates for selected schemes with a spectral-Galerkin approximation are also derived. The stability analyses and error estimates are based on a weak formulation thus the results can be easily extended to other spatial discretizations, such as Galerkin finite element methods, which are based on a weak formulation.