Skip to main content Accessibility help
×
Home
Hostname: page-component-55597f9d44-ms7nj Total loading time: 0.548 Render date: 2022-08-12T15:05:00.049Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "useNewApi": true } hasContentIssue true

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

Published online by Cambridge University Press:  28 May 2015

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: fxlmath@gmail.com
Corresponding author. Email: ttang@math.hkbu.edu.hk
Corresponding author. Email: jiangy@hkbu.edu.hk
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.

Type
Research Article
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) 10851095.CrossRefGoogle 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) 151167.CrossRefGoogle 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) 13671391.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) 913936.CrossRefGoogle 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) 285291.CrossRefGoogle ScholarPubMed
[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) 258267.CrossRefGoogle Scholar
[8]Du, Q., Nicolaides, R.A., Numerical analysis of a continuum model of phase transition, SIAM J. Numer. Anal. 28 (1991) 13101322.CrossRefGoogle 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) 53105327.CrossRefGoogle Scholar
[12]He, Y., Liu, Y., Tang, T., On large time-stepping methods for the Cahn-Hilliard equation, Appl. Numer. Math. 57 (2007) 616628.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) 53235339.CrossRefGoogle Scholar
[14]Li, B., Liu, J.-G., Thin film epitaxy with or without slope selection, European J. Appl. Math. 14(6) (2003) 713743.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) 18931915.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) 13951414.CrossRefGoogle 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) 16691691.CrossRefGoogle 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) 11591179.CrossRefGoogle 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) 22692288.CrossRefGoogle Scholar
[21]Xu, C., Tang, T., Stability analysis of large time-stepping methods for epitaxial growth models, SIAM J. Numer. Anal. 44 (2006) 17591779.CrossRefGoogle Scholar
[22]Yang, X., Error analysis of stabilized semi-implicit method of Allen-Cahn equation, Discrete Contin. Dyn. Syst.-B 11 (2009) 10571070.CrossRefGoogle 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) 30423063.CrossRefGoogle Scholar
[24]Zhang, Z., Qiao, Z., An adaptive time-stepping strategy for the Cahn-Hilliard equation, Commun. Comput. Phys. 11 (2012) 12611278.CrossRefGoogle Scholar
51
Cited by

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

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

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

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

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

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

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *