American Institute of Mathematical Sciences

Advanced
December  2010, 28(4): 1669-1691. doi: 10.3934/dcds.2010.28.1669

Numerical approximations of Allen-Cahn and Cahn-Hilliard equations

Jie Shen 1, and  Xiaofeng Yang 2,

1. 

School of Mathematical Sciences, Xiamen University, Xiamen, 361005, China
2. 

Department of Mathematics, University of South Carolina, Columbia, SC 29208, United States

Received  October 2009 Revised  February 2010 Published  June 2010

Stability analyses and error estimates are carried out for a number of commonly used numerical schemes for the Allen-Cahn and Cahn-Hilliard equations. It is shown that all the schemes we considered are either unconditionally energy stable, or conditionally energy stable with reasonable stability conditions in the semi-discretized versions. Error estimates for selected schemes with a spectral-Galerkin approximation are also derived. The stability analyses and error estimates are based on a weak formulation thus the results can be easily extended to other spatial discretizations, such as Galerkin finite element methods, which are based on a weak formulation.
Keywords: Error Analysis, Spectral Method, Cahn-Hilliard, Allen-Cahn, Stability..
Mathematics Subject Classification: Primary: 65N35, 65M15, 65M70; Secondary: 65Z0.
Citation: Jie Shen, Xiaofeng Yang. Numerical approximations of Allen-Cahn and Cahn-Hilliard equations. Discrete & Continuous Dynamical Systems, 2010, 28 (4) : 1669-1691. doi: 10.3934/dcds.2010.28.1669
[1]

Quan Wang, Dongming Yan. On the stability and transition of the Cahn-Hilliard/Allen-Cahn system. Discrete & Continuous Dynamical Systems - B, 2020, 25 (7) : 2607-2620. doi: 10.3934/dcdsb.2020024
[2]

Georgia Karali, Yuko Nagase. On the existence of solution for a Cahn-Hilliard/Allen-Cahn equation. Discrete & Continuous Dynamical Systems - S, 2014, 7 (1) : 127-137. doi: 10.3934/dcdss.2014.7.127
[3]

Christopher P. Grant. Grain sizes in the discrete Allen-Cahn and Cahn-Hilliard equations. Discrete & Continuous Dynamical Systems, 2001, 7 (1) : 127-146. doi: 10.3934/dcds.2001.7.127
[4]

Alain Miranville, Ramon Quintanilla, Wafa Saoud. Asymptotic behavior of a Cahn-Hilliard/Allen-Cahn system with temperature. Communications on Pure & Applied Analysis, 2020, 19 (4) : 2257-2288. doi: 10.3934/cpaa.2020099
[5]

Shixing Li, Dongming Yan. On the steady state bifurcation of the Cahn-Hilliard/Allen-Cahn system. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3077-3088. doi: 10.3934/dcdsb.2018301
[6]

Alain Miranville, Wafa Saoud, Raafat Talhouk. On the Cahn-Hilliard/Allen-Cahn equations with singular potentials. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3633-3651. doi: 10.3934/dcdsb.2018308
[7]

Xiaofeng Yang. Error analysis of stabilized semi-implicit method of Allen-Cahn equation. Discrete & Continuous Dynamical Systems - B, 2009, 11 (4) : 1057-1070. doi: 10.3934/dcdsb.2009.11.1057
[8]

Changchun Liu, Hui Tang. Existence of periodic solution for a Cahn-Hilliard/Allen-Cahn equation in two space dimensions. Evolution Equations & Control Theory, 2017, 6 (2) : 219-237. doi: 10.3934/eect.2017012
[9]

Irena Pawłow. Thermodynamically consistent Cahn-Hilliard and Allen-Cahn models in elastic solids. Discrete & Continuous Dynamical Systems, 2006, 15 (4) : 1169-1191. doi: 10.3934/dcds.2006.15.1169
[10]

Ahmad Makki, Alain Miranville. Existence of solutions for anisotropic Cahn-Hilliard and Allen-Cahn systems in higher space dimensions. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 759-775. doi: 10.3934/dcdss.2016027
[11]

Cristina Pocci. On singular limit of a nonlinear $p$-order equation related to Cahn-Hilliard and Allen-Cahn evolutions. Evolution Equations & Control Theory, 2013, 2 (3) : 517-530. doi: 10.3934/eect.2013.2.517
[12]

Juan Wen, Yaling He, Yinnian He, Kun Wang. Stabilized finite element methods based on multiscale enrichment for Allen-Cahn and Cahn-Hilliard equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021074
[13]

Peter Howard, Bongsuk Kwon. Spectral analysis for transition front solutions in Cahn-Hilliard systems. Discrete & Continuous Dynamical Systems, 2012, 32 (1) : 125-166. doi: 10.3934/dcds.2012.32.125
[14]

Ken Shirakawa. Stability analysis for two dimensional Allen-Cahn equations associated with crystalline type energies. Conference Publications, 2009, 2009 (Special) : 697-707. doi: 10.3934/proc.2009.2009.697
[15]

Charles-Edouard Bréhier, Ludovic Goudenège. Analysis of some splitting schemes for the stochastic Allen-Cahn equation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4169-4190. doi: 10.3934/dcdsb.2019077
[16]

Hongmei Cheng, Rong Yuan. Multidimensional stability of disturbed pyramidal traveling fronts in the Allen-Cahn equation. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 1015-1029. doi: 10.3934/dcdsb.2015.20.1015
[17]

Xinlong Feng, Huailing Song, Tao Tang, Jiang Yang. Nonlinear stability of the implicit-explicit methods for the Allen-Cahn equation. Inverse Problems & Imaging, 2013, 7 (3) : 679-695. doi: 10.3934/ipi.2013.7.679
[18]

Hirokazu Ninomiya, Masaharu Taniguchi. Global stability of traveling curved fronts in the Allen-Cahn equations. Discrete & Continuous Dynamical Systems, 2006, 15 (3) : 819-832. doi: 10.3934/dcds.2006.15.819
[19]

Pierluigi Colli, Gianni Gilardi, Paolo Podio-Guidugli, Jürgen Sprekels. An asymptotic analysis for a nonstandard Cahn-Hilliard system with viscosity. Discrete & Continuous Dynamical Systems - S, 2013, 6 (2) : 353-368. doi: 10.3934/dcdss.2013.6.353
[20]

Laurence Cherfils, Madalina Petcu, Morgan Pierre. A numerical analysis of the Cahn-Hilliard equation with dynamic boundary conditions. Discrete & Continuous Dynamical Systems, 2010, 27 (4) : 1511-1533. doi: 10.3934/dcds.2010.27.1511

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (684)
  • HTML views (0)
  • Cited by (319)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences