June  2009, 11(4): 1057-1070. doi: 10.3934/dcdsb.2009.11.1057

Error analysis of stabilized semi-implicit method of Allen-Cahn equation

1. 

Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599, United States

Received  July 2008 Revised  October 2008 Published  April 2009

We consider in this paper the stabilized semi-implicit (in time) scheme and the splitting scheme for the Allen-Cahn equation $\phi_t-\Delta\phi+$ε$^-2f(\phi)=0$ arising from phase transitions in material science. For the stabilized first-order scheme, we show that it is unconditionally stable and the error bound depends on ε-1 in some lower polynomial order using the spectrum estimate of [2, 10, 11]. In addition, the first- and second-order operator splitting schemes are proposed and the accuracy are tested and compared with the semi-implicit schemes numerically.
Citation: Xiaofeng Yang. Error analysis of stabilized semi-implicit method of Allen-Cahn equation. Discrete and Continuous Dynamical Systems - B, 2009, 11 (4) : 1057-1070. doi: 10.3934/dcdsb.2009.11.1057
[1]

Xufeng Xiao, Xinlong Feng, Jinyun Yuan. The stabilized semi-implicit finite element method for the surface Allen-Cahn equation. Discrete and Continuous Dynamical Systems - B, 2017, 22 (7) : 2857-2877. doi: 10.3934/dcdsb.2017154

[2]

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

[3]

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

[4]

Gianni Gilardi. On an Allen-Cahn type integrodifferential equation. Discrete and Continuous Dynamical Systems - S, 2013, 6 (3) : 703-709. doi: 10.3934/dcdss.2013.6.703

[5]

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

[6]

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

[7]

Murat Uzunca, Ayşe Sarıaydın-Filibelioǧlu. Adaptive discontinuous galerkin finite elements for advective Allen-Cahn equation. Numerical Algebra, Control and Optimization, 2021, 11 (2) : 269-281. doi: 10.3934/naco.2020025

[8]

Tatsuki Mori, Kousuke Kuto, Tohru Tsujikawa, Shoji Yotsutani. Representation formulas of solutions and bifurcation sheets to a nonlocal Allen-Cahn equation. Discrete and Continuous Dynamical Systems, 2020, 40 (8) : 4907-4925. doi: 10.3934/dcds.2020205

[9]

Christos Sourdis. On the growth of the energy of entire solutions to the vector Allen-Cahn equation. Communications on Pure and Applied Analysis, 2015, 14 (2) : 577-584. doi: 10.3934/cpaa.2015.14.577

[10]

Paul H. Rabinowitz, Ed Stredulinsky. On a class of infinite transition solutions for an Allen-Cahn model equation. Discrete and Continuous Dynamical Systems, 2008, 21 (1) : 319-332. doi: 10.3934/dcds.2008.21.319

[11]

Ciprian G. Gal, Maurizio Grasselli. The non-isothermal Allen-Cahn equation with dynamic boundary conditions. Discrete and Continuous Dynamical Systems, 2008, 22 (4) : 1009-1040. doi: 10.3934/dcds.2008.22.1009

[12]

Eleonora Cinti. Saddle-shaped solutions for the fractional Allen-Cahn equation. Discrete and Continuous Dynamical Systems - S, 2018, 11 (3) : 441-463. doi: 10.3934/dcdss.2018024

[13]

Zhuoran Du, Baishun Lai. Transition layers for an inhomogeneous Allen-Cahn equation in Riemannian manifolds. Discrete and Continuous Dynamical Systems, 2013, 33 (4) : 1407-1429. doi: 10.3934/dcds.2013.33.1407

[14]

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 and Applied Analysis, 2022, 21 (6) : 1873-1894. doi: 10.3934/cpaa.2021074

[15]

Hector D. Ceniceros. A semi-implicit moving mesh method for the focusing nonlinear Schrödinger equation. Communications on Pure and Applied Analysis, 2002, 1 (1) : 1-18. doi: 10.3934/cpaa.2002.1.1

[16]

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

[17]

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

[18]

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

[19]

Suting Wei, Jun Yang. Clustering phase transition layers with boundary intersection for an inhomogeneous Allen-Cahn equation. Communications on Pure and Applied Analysis, 2020, 19 (5) : 2575-2616. doi: 10.3934/cpaa.2020113

[20]

Fang Li, Kimie Nakashima. Transition layers for a spatially inhomogeneous Allen-Cahn equation in multi-dimensional domains. Discrete and Continuous Dynamical Systems, 2012, 32 (4) : 1391-1420. doi: 10.3934/dcds.2012.32.1391

2021 Impact Factor: 1.497

Metrics

  • PDF downloads (209)
  • HTML views (0)
  • Cited by (38)

Other articles
by authors

[Back to Top]