Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-23T12:16:27.206Z Has data issue: false hasContentIssue false
  • English
  • Français

Stabilized Crank-Nicolson/Adams-Bashforth Schemes for Phase Field Models

Published online by Cambridge University Press:  28 May 2015

Xinlong Feng ,
Tao Tang  and
Jiang Yang
Xinlong Feng*
Affiliation:
Department of Mathematics & Institute for Computational and Theoretical Studies, Hong Kong Baptist University, Kowloon Tong, Hong Kong, China College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, China
Tao Tang*
Affiliation:
Department of Mathematics & Institute for Computational and Theoretical Studies, Hong Kong Baptist University, Kowloon Tong, Hong Kong, China
Jiang Yang*
Affiliation:
Department of Mathematics & Institute for Computational and Theoretical Studies, Hong Kong Baptist University, Kowloon Tong, Hong Kong, China
*
Corresponding author. Email: [email protected]
Corresponding author. Email: [email protected]
Corresponding author. Email: [email protected]
Get access

Abstract

In this paper, stabilized Crank-Nicolson/Adams-Bashforth schemes are presented for the Allen-Cahn and Cahn-Hilliard equations. It is shown that the proposed time discretization schemes are either unconditionally energy stable, or conditionally energy stable under some reasonable stability conditions. Optimal error estimates for the semi-discrete schemes and fully-discrete schemes will be derived. Numerical experiments are carried out to demonstrate the theoretical results.

Keywords

35L7065N3076D06Allen-Cahn equationCahn-Hilliard equationCrank-Nicolson schemeAdams-Bashforth schemeimplicit-explicit methoderror estimates
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 3 , Issue 1 , February 2013 , pp. 59 - 80
Copyright
Copyright © Global-Science Press 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Allen, S.M., Cahn, J.W., A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metall. 27 (1979) 1085–1095.Google Scholar
[2]Ascher, Uri M., Ruuth, J., Spiteri, R.J., Implicit-Explicit Runge-Kutta method for time dependent partial differential equations, Appl. Numer. Math. 25 (1997) 151–167.Google Scholar
[3]Bertozzi, A.L., Ju, N., Lu, H.-W., A biharmonic modified forward time stepping method for fourth order nonlinear diffusion equations, Discrete Contin. Dyn. Syst. 29(4) (2011) 1367–1391.Google Scholar
[4]Bertozzi, A.L., Esedoglu, S., Gillette, A., Analysis of a two-scale Cahn-Hilliard model for image inpainting, Multi. Model. Simul. 6(3) (2007) 913–936.Google Scholar
[5]Bertozzi, A.L., Esedoglu, S., Gillette, A., Inpainting of binary images using the Cahn-Hilliard equation, IEEE Trans. Image Proc. 16(1) (2007) 285–291.Google Scholar
[6]Boscarino, S., Pareschi, L., Russo, G., Implicit-Explicit Runge-Kutta schemes for hyperbolic systems and kinetic equations in the diffusion limit, SIAM J. Sci. Comp. to appear.Google Scholar
[7]Cahn, J.W., Hilliard, J.E., Free energy of a nonuniform system, I: Interfacial free energy, J. Chem. Phys. 28(2) (1958) 258–267.Google Scholar
[8]Du, Q., Nicolaides, R.A., Numerical analysis of a continuum model of phase transition, SIAM J. Numer. Anal. 28 (1991) 1310–1322.Google Scholar
[9]Eyre, D.J., An unconditionally stable one-step scheme for gradient systems, unpublished, http://www.math.utah.edu/eyre/research/methods/stable.ps.Google Scholar
[10]Feng, X., Song, H., Tang, T., Yang, J., Nonlinearly stable implicit-explicit methods for the Allen-Cahn equation, Preprint.Google Scholar
[11]Gomez, H., Hughes, T.J.R., Provably unconditionally stable, second-order time-accurate, mixed variational methods for phase-field models, J. Comput. Phys. 230(13) (2011) 5310–5327.Google Scholar
[12]He, Y., Liu, Y., Tang, T., On large time-stepping methods for the Cahn-Hilliard equation, Appl. Numer. Math. 57 (2007) 616–628.CrossRefGoogle Scholar
[13]Hu, Z., Wise, S.M., Wang, C., Lowengrub, J.S., Stable and efficient finite-diffence nonlinear multigrid schemes for the phase field crystal equation, J. Comput. Phys. 228 (2009) 5323–5339.Google Scholar
[14]Li, B., Liu, J.-G., Thin film epitaxy with or without slope selection, European J. Appl. Math. 14(6) (2003) 713–743.CrossRefGoogle Scholar
[15]Qiao, Z., Sun, Z., Zhang, Z., The stability and convergence of two linearized finite difference schemes for the nonlinear epitaxial growth model, Numer. Methods Part. Diff. Eq. 28(6) (2012) 1893–1915.CrossRefGoogle Scholar
[16]Qiao, Z., Zhang, Z., Tang, T., An adaptive time-stepping strategy for the molecular beam epitaxy models, SIAM J. Sci. Comput. 33 (2011) 1395–1414.Google Scholar
[17]Shen, J., Tang, T., Wang, L., Spectral Methods: Algorithms, Analysis and Applications, Volume 41 of Springer Series in Computational Mathematics, Springer, 2011.Google Scholar
[18]Shen, J., Yang, X., Numerical approximations of Allen-Cahn and Cahn-Hilliard equations, Discrete Contin. Dyn. Syst.-A 28 (2010) 1669–1691.Google Scholar
[19]Shen, J., Yang, X., A phase-field model and its numerical approximation for two-phase incompressible flows with different densities and viscosities, SIAM J. Sci. Comput. 32 (2010) 1159–1179.Google Scholar
[20]Wise, S.M., Wang, C., Lowengrub, J.S., An energy stable and convergent finite difference scheme for the phase field crystal equation, SIAM J. Numer. Anal. 47 (2009) 2269–2288.Google Scholar
[21]Xu, C., Tang, T., Stability analysis of large time-stepping methods for epitaxial growth models, SIAM J. Numer. Anal. 44 (2006) 1759–1779.Google Scholar
[22]Yang, X., Error analysis of stabilized semi-implicit method of Allen-Cahn equation, Discrete Contin. Dyn. Syst.-B 11 (2009) 1057–1070.Google Scholar
[23]Zhang, J., Du, Q., Numerical studies of discrete approximations to the Allen-Cahn equation in the sharp interface limit, SIAM J. Sci. Comput. 31(4) (2009) 3042–3063.CrossRefGoogle Scholar
[24]Zhang, Z., Qiao, Z., An adaptive time-stepping strategy for the Cahn-Hilliard equation, Commun. Comput. Phys. 11 (2012) 1261–1278.Google Scholar