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On a Large Time-Stepping Method for the Swift-Hohenberg Equation

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  19 September 2016

Zhengru Zhang  and
Yuanzi Ma
Zhengru Zhang*
Affiliation:
Laboratory of Mathematics and Complex Systems, Ministry of Education and School of Mathematical Sciences, Beijing Normal University, 19 Xinjiekouwai Street, Haidian District, Beijing 100875, China
Yuanzi Ma*
Affiliation:
College of Science, China University of Petroleum–Beijing, 18 Fuxue Road, Changping, Beijing 102249, China
*
*Corresponding author. Email:zrzhang@bnu.edu.cn (Z. R. Zhang), yzma9017@126.com (Y. Z. Ma)
*Corresponding author. Email:zrzhang@bnu.edu.cn (Z. R. Zhang), yzma9017@126.com (Y. Z. Ma)
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Abstract

The main purpose of this work is to contrast and analyze a large time-stepping numerical method for the Swift-Hohenberg (SH) equation. This model requires very large time simulation to reach steady state, so developing a large time step algorithm becomes necessary to improve the computational efficiency. In this paper, a semi-implicit Euler schemes in time is adopted. An extra artificial term is added to the discretized system in order to preserve the energy stability unconditionally. The stability property is proved rigorously based on an energy approach. Numerical experiments are used to demonstrate the effectiveness of the large time-stepping approaches by comparing with the classical scheme.

Keywords

Large time-stepping methodenergy stableSwift-Hohenberg equationfinite difference method

MSC classification

Secondary: 65M06: Finite difference methods 65Z05: Applications to physics
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 8 , Issue 6 , December 2016 , pp. 992 - 1003
Copyright
Copyright © Global-Science Press 2016 

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References

[1] Allen, S. M. and Cahn, J. W., A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metall., 27 (1979), pp. 1085–1095.Google Scholar
[2] Cahn, J. W. and Hilliard, J. E., Free energy of a nonuniform system I: Interface free energy, J. Chem. Phys., 28 (1958), pp. 258–267.Google Scholar
[3] Ceniceros, H. D., Roma, A. M., Silveira-Neto, A. and Villar, M. M., A robust, fully adaptive hybrid level-set/ front-tracking method for two-phase flows with an accurate surface tension computation, Commun. Comput. Phys., 8 (2010), pp. 51–94.Google Scholar
[4] Cheng, M.W. and Warren, J. A., An efficient algorithmfor solving the phase field crystal model, J. Comput. Phys., 227 (2008), pp. 6241–6248.CrossRefGoogle Scholar
[5] Elder, K. R., Katakowski, M., Haataja, M. and Grant, M., Modeling elasticity in crystal growth, Phys. Rev. Lett., 88 (2002), 245701.Google Scholar
[6] He, Y., Liu, Y., and Tang, T., On large time-stepping methods for the Cahn-Hilliard equation, Appl. Numer. Math., 57 (2007), pp. 616–628.CrossRefGoogle Scholar
[7] Qiao, Z. H., Sun, Z. Z. and Zhang, Z. R., The stability and convergence of two linearized finite difference schemes for the nonlinear epitaxial growth model, Numerical Methods for Partial Differential Equations, 28 (2012), pp. 1893–1915.Google Scholar
[8] Qiao, Z. H., Zhang, Z. R. and Tang, T., An adaptive time-stepping strategy for the molecular beam epitaxy models, SIAM J. Sci. Comput., 33 (2011), pp. 1395–1414.CrossRefGoogle Scholar
[9] Rogers, T. M., Elder, K. R. and Desai, R. C., Numerical study of the late stages of spinodal decomposition, Phys. Rev., B37 (1988), pp. 9638–9649.Google Scholar
[10] Swift, J. and Hohenberg, P. C., Hydrodynamic fluctuations at the convective instability, Phys. Rev. A, 15 (1977), pp. 319–328.Google Scholar
[11] Tsao, J. Y., Materials Fundamentals of Molecular Beam Epitaxy, World-Scientific, Singapore, 1993.Google Scholar
[12] Wise, S. M., Wang, C. and Lowengrub, J. S., An energy-stable and convergent finite-difference scheme for the phase field crystal wquation, SIAM J. Number. Anal., 47 (2009), pp. 2269–2288.CrossRefGoogle Scholar
[13] Xu, C. J. and Tang, T., Stability analysis of large time-stepping methods for epitaxial growth models, SIAM J. Numer. Anal., 44 (2006), pp. 1759–1779.Google Scholar
[14] Zhang, S. P., Liu, C. and Zhang, H., Numerical simulations of hydrodynamics of nematic liquid crystals: effects of kinematic transports, Commun. Comput. Phys., 9 (2011), pp. 974–993.Google Scholar
[15] Zhang, Z. R. and Qiao, Z. H., An adaptive time-stepping strategy for Cahn-Hilliard equation, Commun. Comput. Phys., 11 (2012), pp. 1261–1278.CrossRefGoogle Scholar