粘性Cahn-Hilliard方程Crank-Nicolson/Adams-Bashforth格式的混合有限元方法
Crank-Nicolson/Adams-Bashforth Scheme of Mixed Finite Element Method for the Viscous Cahn-Hilliard Equation
DOI: 10.12677/AAM.2020.98135, PDF,    国家自然科学基金支持
作者: 卫钱瑞, 牛丽芳：太原理工大学数学学院，山西 晋中
关键词: 粘性Cahn-Hilliard方程；Crank-Nicolson；Adams-Bashforth；稳定性；误差估计；Viscous Cahn-Hilliard Equation； Crank-Nicolson； Adams-Bashforth； Stability； Error Estimates
摘要: 本文基于线性的Crank-Nicolson/Adams-Bashforth格式，建立了有效稳定的数值算法求解粘性Cahn-Hilliard方程。在该算法中，非线性体积力被显示处理，导致求解具有常系数的线性系统，从而提高算法效率。此外，该算法的稳定性以及误差估计被详细证明。最终，通过数值实验证明了该格式的稳定性及收敛阶。
Abstract: In this paper, an efficient stabilized algorithm based on the linear Crank-Nicolson/Adams-Bashforth scheme is proposed to solve the viscous Cahn-Hilliard equation. In this algorithm, the nonlinear bulk force is treated explicitly with linear stabilization term. This treatment leads to solve linear systems with constant coefficients that can improve algorithm efficiency. Further, the stability analysis and priori error estimates on proposed method are provided in detail. Finally, a series of numerical experiments are implemented to illustrate the theoretical analysis.
文章引用：卫钱瑞, 牛丽芳. 粘性Cahn-Hilliard方程Crank-Nicolson/Adams-Bashforth格式的混合有限元方法[J]. 应用数学进展, 2020, 9(8): 1159-1169. https://doi.org/10.12677/AAM.2020.98135

参考文献

[1] Cahn, J.W. and Hilliard, J.E. (1958) Free Energy of a Nonuniform System. I. Interfacial Free Energy. Journal of Chemical Physics, 28, 258-267. [Google Scholar] [CrossRef] 
[2] Novick-Cohen, A. (1988) On the Viscous Chan-Hilliard Equation in Material Instabilities in Continuum and Related Mathemati-Cal Problems. Oxford University Press, Oxford.
[3] Novick-Cohen, P. (1991) Stable Patterns in a Viscous Diffusion Equation. Transactions of the American Mathematical Society, 324, 331-351. [Google Scholar] [CrossRef] 
[4] Ascher, U.M., Ruuth, S.J., Spiteri, R.J., et al. (1997) Implicit-Explicit Runge-Kutta Methods for Time-Dependent Partial Differential Equations. Applied Numerical Mathematics, 25, 151-167. [Google Scholar] [CrossRef] 
[5] Boscarino, S., Pareschi, L. and Russo, G. (2013) Implicit-Explicit Runge-Kutta Schemes for Hyperbolic Systems and Kinetic Equations in the Diffusion Limit. Siam Journal on Scientific Computing, 35, A22-A51. [Google Scholar] [CrossRef] 
[6] Shen, J. and Tang, T. (2011) Spectral Methods: Algorithms, Analysis and Applications. Springer Publishing Company, Incorporated. [Google Scholar] [CrossRef] 
[7] Yang, J., Tang, T., Song, H., et al. (2013) Nonlinear Stability of the Implicit-Explicit Methods for the Allen-Cahn Equation. Inverse Problems & Imaging, 7, 679-695. [Google Scholar] [CrossRef] 
[8] Feng, X., Tang, T. and Yang, J. (2013) Stabilized Crank-Nicolson/Adams-Bashforth Schemes for Phase Field Models. East Asian Journal on Applied Mathematics, 3, 59-80. [Google Scholar] [CrossRef] 
[9] Yu, L. (2018) Convergence Analysis of an Unconditinally Energy Stable Linear Crank-Nicolson Scheme for the Cahn-Hilliard Equation. The Journal of Men’s Studies, 51, 89-114. [Google Scholar] [CrossRef] 
[10] Yan, Y., Chen, W., Wang, C. and Wise, S.M. (2018) A Second-Order Energy Stable BDF Numerical Scheme for the Cahn-Hilliard Equation. Communications in Computational Physics, 23, 572-602. [Google Scholar] [CrossRef] 
[11] Shen, J. and Yang, X. (2010) Numerical Approximations of Allen-Cahn and Cahn-Hilliard Equations. Discrete and Continuous Dynamical Systems, 28, 1669-1691. [Google Scholar] [CrossRef] 

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