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An Adaptive Time-Stepping Strategy for the Cahn-Hilliard Equation

Published online by Cambridge University Press:  20 August 2015

Zhengru Zhang  and
Zhonghua Qiao
Zhengru Zhang*
Affiliation:
Laboratory of Mathematics and Complex Systems, Ministry of Education; School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China
Zhonghua Qiao*
Affiliation:
Institute for Computational Mathematics & Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong
*
Email:zrzhang@bnu.edu.cn
Corresponding author.Email:zqiao@hkbu.edu.hk
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Abstract

This paper studies the numerical simulations for the Cahn-Hilliard equation which describes a phase separation phenomenon. The numerical simulation of the Cahn-Hilliard model needs very long time to reach the steady state, and therefore large time-stepping methods become useful. The main objective of this work is to construct the unconditionally energy stable finite difference scheme so that the large time steps can be used in the numerical simulations. The equation is discretized by the central difference scheme in space and fully implicit second-order scheme in time. The proposed scheme is proved to be unconditionally energy stable and mass-conservative. An error estimate for the numerical solution is also obtained with second order in both space and time. By using this energy stable scheme, an adaptive time-stepping strategy is proposed, which selects time steps adaptively based on the variation of the free energy against time. The numerical experiments are presented to demonstrate the effectiveness of the adaptive time-stepping approach.

Keywords

35L6465M9365M30Adaptive time-steppingunconditionally energy stableCahn-Hilliard equationmass conservation
Type
Research Article
Information
Communications in Computational Physics , Volume 11 , Issue 4 , April 2012 , pp. 1261 - 1278
Copyright
Copyright © Global Science Press Limited 2012

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References

[1]Cahn, J. W. and Hilliard, J. E., Free energy of a non-uniform system I: interfacial free energy, J. Chem. Phys., 28 (1958), 258–267.Google Scholar
[2]Carr, J., Curtin, M. E. and Slemrod, M., Structured phase transitions on a finite interval, Arch. Rat. Mech. Anal., 86 (1984), 317–351.CrossRefGoogle Scholar
[3]Cheng, M. W. and Warren, J. A., Controlling the accuracy of unconditionally stable algorithms in the Cahn-Hilliard equation, Phys. Rev. E, 75 (2007), 017702.CrossRefGoogle ScholarPubMed
[4]Choo, S. M., Chung, S. K. and Kim, K. I., Conservative nonlinear difference scheme for the Cahn-Hilliard equation-II, Comput. Math. Appl., 39 (2000), 229–243.Google Scholar
[5]Du, Q. and Nicolaides, R. A., Numerical analysis of a continuum model of phase transition, SIAM J. Numer. Anal., 28 (1994), 1310–1322.Google Scholar
[6]Elliott, C. M. and Sonqmu, Z., On the Cahn-Hilliard equation, Arch. Rat. Meth. Anal., 96 (1986), 339–357.CrossRefGoogle Scholar
[7]Elliott, C. M. and French, D. A., Numerical studies of the Cahn-Hilliard equation for phase separation, IMA J. Appl. Math., 38 (1987), 97–128.Google Scholar
[8]Elliott, C. M. and French, D. A., A nonconforming finite element method for the two-dimensional Cahn-Hilliard equations, SIAM J. Numer. Anal., 26 (1989), 884–903.Google Scholar
[9]Elliott, C. M. and Larsson, S., Error estimates with smooth and nonsmooth data for a finite element method for the Cahn-Hilliard equation, Math. Comput., 58 (1992), 603–630.Google Scholar
[10]Furihata, D., A stable and conservative finite difference scheme for the Cahn-Hilliard equation, Numer. Math., 87 (2001), 675–699.Google Scholar
[11]He, Y. N., Liu, Y. X. and Tang, T., On large time-stepping methods for the Cahn-Hilliard equation, Appl. Numer. Math., 57 (2007), 616–628.Google Scholar
[12]Liu, C. and Shen, J., A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method, Phys. D, 179 (2003), 211–228.Google Scholar
[13]Minkoff, S. E. and Kridler, N. M., A comparison of adaptive time stepping methods for coupled flow and deformation modeling, Appl. Math. Model., 30 (2006), 993–1009.CrossRefGoogle Scholar
[14] A. Novick-Cohen and Segel, L. A., Nonlinear aspects of the Cahn-Hilliard equation, Phys. D, 10 (1984), 277–298.Google Scholar
[15]Qiao, Z. H., Zhang, Z. R. and Tang, T., An adaptive time-stepping strategy for the epitaxial growth models, SIAM J. Sci. Comput., 33 (2011), 1395–1414.CrossRefGoogle Scholar
[16]Söderlind, G., Automatic control and adptive time-stepping, Numer. Algor., 31 (2001), 281–310.CrossRefGoogle Scholar
[17]Sun, Z. Z., A second-order accurate linearized difference scheme for the two-dimensional Cahn-Hilliard equation, Math. Comput., 64 (1995), 1463–1471.Google Scholar
[18]Tan, Z. J., Zhang, Z. R., Tang, T. and Huang, Y. Q., Moving mesh methods with locally varying time steps, J. Comput. Phys., 200 (2004), 347–367.CrossRefGoogle Scholar
[19]Wise, S. M., Unconditionally stable finite difference, nonlinear multigrid simulation of the Cahn-Hilliard-Hele-Shaw system of equations, J. Sci. Comput., 44 (2010), 38–68.Google Scholar
[20]Xu, C. J. and Tang, T., Stability analysis of large time-stepping methods for epitaxial growth models, SIAM J. Numer. Anal., 44 (2006), 1759–1779.Google Scholar
[21]Zhang, S. and Wang, M., A nonconforming finite element method for Cahn-Hilliard equation, J. Comput. Phys., 229 (2010), 7361–7372.Google Scholar
[22]Zhou, Y. L., Applications of Discrete Functional Analysis to the Finite Difference Method, International Academic Publishers, 1990.Google Scholar