August  2013, 7(3): 679-695. doi: 10.3934/ipi.2013.7.679

Nonlinear stability of the implicit-explicit methods for the Allen-Cahn equation

1. 

College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, China

2. 

Institute of Theoretical and Computational Studies & Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong, China, China, China

Received  November 2012 Revised  June 2013 Published  September 2013

In this paper, we will investigate the first- and second-order implicit-explicit schemes with parameters for solving the Allen-Cahn equation. It is known that the Allen-Cahn equation satisfies a nonlinear stability property, i.e., the free-energy functional decreases in time. The goal of this paper is to consider implicit-explicit schemes that inherit the nonlinear stability of the continuous model, which will be achieved by properly choosing parameters associated with the implicit-explicit schemes. Theoretical justifications for the nonlinear stability of the schemes will be provided, and the theoretical results will be verified by several numerical examples.
Citation: 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
References:
[1]

S. M. Allen and J. W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening,, Acta Metall, 27 (1979), 1085. doi: 10.1016/0001-6160(79)90196-2.

[2]

U. M. Ascher, J. Ruuth and R. J. Spiteri, Implicit-explicit Runge-Kutta method for time dependent partial differential equations,, Special issue on time integration (Amsterdam, 25 (1997), 151. doi: 10.1016/S0168-9274(97)00056-1.

[3]

U. M. Ascher, J. Ruuth and T. R. Wetton, Implicit-explicit method for time dependent partial differential equations,, SIAM J. Numer. Anal., 32 (1995), 797. doi: 10.1137/0732037.

[4]

A. L. Bertozzi, N. Ju and H.-W. Lu, A biharmonic modified forward time stepping method for fourth order nonlinear diffusion equations,, Discrete Contin. Dyn. Syst. A, 29 (2011), 1367.

[5]

A. L. Bertozzi, S. Esedoglu and A. Gillette, Analysis of a two-scale Cahn-Hilliard model for image inpainting,, Multiscale Model. Simul., 6 (2007), 913. doi: 10.1137/060660631.

[6]

A. L. Bertozzi, S. Esedoglu and A. Gillette, Inpainting of binary images using the Cahn-Hilliard equation,, IEEE Trans. Image Proc., 16 (2007), 285. doi: 10.1109/TIP.2006.887728.

[7]

S. Boscarino, L. Pareschi and G. Russo, Implicit-explicit Runge-Kutta schemes for hyperbolic systems and kinetic equations in the diffusion limit,, SIAM J. Sci. Comput., 35 (2013). doi: 10.1137/110842855.

[8]

F. Chen and J. Shen, Efficient energy stable schemes with spectral discretization in space for anisotropic Cahn-Hilliard systems,, Commun. Comput. Phys., 13 (2013), 1189.

[9]

C. Collins, J. Shen and S. M. Wise, An efficient, energy stable scheme for the Cahn-Hilliard-Brinkman system,, Commun. Comput. Phys., 13 (2013), 929.

[10]

Charles M. Elliott and Bjorn Stinner, Computation of two-phase biomembranes with phase dependent material parameters using surface finite elements,, Commun. Comput. Phys., 13 (2013), 325.

[11]

D. J. Eyre, An unconditionally stable one-step scheme for gradient systems, unpublished article, 1998., , ().

[12]

X. Feng, T. Tang and J. Yang, Stabilized Crank-Nicolson/Adams-Bashforth schemes for phase field models,, East Asian J. Appl. Math., 3 (2013), 59. doi: 10.4208/eajam.200113.220213a.

[13]

L. Golubovic, A. Levandovsky and D. Moldovan, Interface dynamics and far-from-equilibrium phase transitions in multilayer epitaxial growth and erosion on crystal surfaces: Continuum theory insights,, East Asian J. Appl. Math., 1 (2011), 297. doi: 10.4208/eajam.040411.030611a.

[14]

F. de la Hoz and F. Vadillo, A Sylvester-based IMEX method via differentiation matrices for solving nonlinear parabolic equations,, Commun. Comput. Phys., 14 (2013), 1001.

[15]

Y. He, Y. Liu and T. Tang, On large time-stepping methods for the Cahn-Hilliard equation,, Appl. Numer. Math., 57 (2007), 616. doi: 10.1016/j.apnum.2006.07.026.

[16]

Samet Y. Kadioglu and Dana A. Knoll, A Jacobian-free Newton Krylov implicit-explicit time integration method for incompressible flow problems,, Commun. Comput. Phys., 13 (2013), 1408.

[17]

A. Kassam and L. N. Trefethen, Fourth-order time-stepping for stiff PDEs,, SIAM J. Sci. Comput., 26 (2005), 1214. doi: 10.1137/S1064827502410633.

[18]

Junseok Kim, Phase-field models for multi-component fluid flows,, Commun. Comput. Phys., 12 (2012), 613. doi: 10.4208/cicp.301110.040811a.

[19]

M. Li, T. Tang and B. Fornberg, A compact fourth-order finite difference scheme for the steady incompressible Navier-Stokes equations,, Int. J. Numer. Methods Fluids, 20 (1995), 1137. doi: 10.1002/fld.1650201003.

[20]

M. Li and T. Tang, A compact fourth-order finite difference scheme for unsteady viscous incompressible flows,, J. Sci. Comp., 16 (2001), 29. doi: 10.1023/A:1011146429794.

[21]

B. Li and J.-G. Liu, Thin film epitaxy with or without slope selection,, European J. Appl. Math., 14 (2003), 713. doi: 10.1017/S095679250300528X.

[22]

Z. Qiao, Z. Sun and Z. Zhang, The stability and convergence of two linearized finite difference schemes for the nonlinear epitaxial growth model,, Numer. Meth. Part Differ. Equ., 28 (2012), 1893. doi: 10.1002/num.20707.

[23]

Z. Qiao, Z. Zhang and T. Tang, An adaptive time-stepping strategy for the molecular beam epitaxy models,, SIAM J. Sci. Comput., 33 (2011), 1395. doi: 10.1137/100812781.

[24]

J. Shen, T. Tang and L. Wang, "Spectral Methods: Algorithms, Analysis and Applications,", 41 Springer, 41 (2011). doi: 10.1007/978-3-540-71041-7.

[25]

J. Shen and X. Yang, Numerical approximations of Allen-Cahn and Cahn-Hilliard equations,, Discrete Contin. Dyn. Syst. A, 28 (2010), 1669. doi: 10.3934/dcds.2010.28.1669.

[26]

C. Xu and T. Tang, Stability analysis of large time-stepping methods for epitaxial growth models,, SIAM J. Numer. Anal., 44 (2006), 1759. doi: 10.1137/050628143.

[27]

X. Yang, Error analysis of stabilized semi-implicit method of Allen-Cahn equation,, Discrete Contin. Dyn. Syst. Ser. B, 11 (2009), 1057. doi: 10.3934/dcdsb.2009.11.1057.

[28]

J. Zhang and Q. Du, Numerical studies of discrete approximations to the Allen-Cahn equation in the sharp interface limit,, SIAM J. Sci. Comput., 31 (2009), 3042. doi: 10.1137/080738398.

[29]

Z. Zhang and Z. Qiao, An adaptive time-stepping strategy for the Cahn-Hilliard equation,, Commun. Comput. Phys., 11 (2012), 1261. doi: 10.4208/cicp.300810.140411s.

show all references

References:
[1]

S. M. Allen and J. W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening,, Acta Metall, 27 (1979), 1085. doi: 10.1016/0001-6160(79)90196-2.

[2]

U. M. Ascher, J. Ruuth and R. J. Spiteri, Implicit-explicit Runge-Kutta method for time dependent partial differential equations,, Special issue on time integration (Amsterdam, 25 (1997), 151. doi: 10.1016/S0168-9274(97)00056-1.

[3]

U. M. Ascher, J. Ruuth and T. R. Wetton, Implicit-explicit method for time dependent partial differential equations,, SIAM J. Numer. Anal., 32 (1995), 797. doi: 10.1137/0732037.

[4]

A. L. Bertozzi, N. Ju and H.-W. Lu, A biharmonic modified forward time stepping method for fourth order nonlinear diffusion equations,, Discrete Contin. Dyn. Syst. A, 29 (2011), 1367.

[5]

A. L. Bertozzi, S. Esedoglu and A. Gillette, Analysis of a two-scale Cahn-Hilliard model for image inpainting,, Multiscale Model. Simul., 6 (2007), 913. doi: 10.1137/060660631.

[6]

A. L. Bertozzi, S. Esedoglu and A. Gillette, Inpainting of binary images using the Cahn-Hilliard equation,, IEEE Trans. Image Proc., 16 (2007), 285. doi: 10.1109/TIP.2006.887728.

[7]

S. Boscarino, L. Pareschi and G. Russo, Implicit-explicit Runge-Kutta schemes for hyperbolic systems and kinetic equations in the diffusion limit,, SIAM J. Sci. Comput., 35 (2013). doi: 10.1137/110842855.

[8]

F. Chen and J. Shen, Efficient energy stable schemes with spectral discretization in space for anisotropic Cahn-Hilliard systems,, Commun. Comput. Phys., 13 (2013), 1189.

[9]

C. Collins, J. Shen and S. M. Wise, An efficient, energy stable scheme for the Cahn-Hilliard-Brinkman system,, Commun. Comput. Phys., 13 (2013), 929.

[10]

Charles M. Elliott and Bjorn Stinner, Computation of two-phase biomembranes with phase dependent material parameters using surface finite elements,, Commun. Comput. Phys., 13 (2013), 325.

[11]

D. J. Eyre, An unconditionally stable one-step scheme for gradient systems, unpublished article, 1998., , ().

[12]

X. Feng, T. Tang and J. Yang, Stabilized Crank-Nicolson/Adams-Bashforth schemes for phase field models,, East Asian J. Appl. Math., 3 (2013), 59. doi: 10.4208/eajam.200113.220213a.

[13]

L. Golubovic, A. Levandovsky and D. Moldovan, Interface dynamics and far-from-equilibrium phase transitions in multilayer epitaxial growth and erosion on crystal surfaces: Continuum theory insights,, East Asian J. Appl. Math., 1 (2011), 297. doi: 10.4208/eajam.040411.030611a.

[14]

F. de la Hoz and F. Vadillo, A Sylvester-based IMEX method via differentiation matrices for solving nonlinear parabolic equations,, Commun. Comput. Phys., 14 (2013), 1001.

[15]

Y. He, Y. Liu and T. Tang, On large time-stepping methods for the Cahn-Hilliard equation,, Appl. Numer. Math., 57 (2007), 616. doi: 10.1016/j.apnum.2006.07.026.

[16]

Samet Y. Kadioglu and Dana A. Knoll, A Jacobian-free Newton Krylov implicit-explicit time integration method for incompressible flow problems,, Commun. Comput. Phys., 13 (2013), 1408.

[17]

A. Kassam and L. N. Trefethen, Fourth-order time-stepping for stiff PDEs,, SIAM J. Sci. Comput., 26 (2005), 1214. doi: 10.1137/S1064827502410633.

[18]

Junseok Kim, Phase-field models for multi-component fluid flows,, Commun. Comput. Phys., 12 (2012), 613. doi: 10.4208/cicp.301110.040811a.

[19]

M. Li, T. Tang and B. Fornberg, A compact fourth-order finite difference scheme for the steady incompressible Navier-Stokes equations,, Int. J. Numer. Methods Fluids, 20 (1995), 1137. doi: 10.1002/fld.1650201003.

[20]

M. Li and T. Tang, A compact fourth-order finite difference scheme for unsteady viscous incompressible flows,, J. Sci. Comp., 16 (2001), 29. doi: 10.1023/A:1011146429794.

[21]

B. Li and J.-G. Liu, Thin film epitaxy with or without slope selection,, European J. Appl. Math., 14 (2003), 713. doi: 10.1017/S095679250300528X.

[22]

Z. Qiao, Z. Sun and Z. Zhang, The stability and convergence of two linearized finite difference schemes for the nonlinear epitaxial growth model,, Numer. Meth. Part Differ. Equ., 28 (2012), 1893. doi: 10.1002/num.20707.

[23]

Z. Qiao, Z. Zhang and T. Tang, An adaptive time-stepping strategy for the molecular beam epitaxy models,, SIAM J. Sci. Comput., 33 (2011), 1395. doi: 10.1137/100812781.

[24]

J. Shen, T. Tang and L. Wang, "Spectral Methods: Algorithms, Analysis and Applications,", 41 Springer, 41 (2011). doi: 10.1007/978-3-540-71041-7.

[25]

J. Shen and X. Yang, Numerical approximations of Allen-Cahn and Cahn-Hilliard equations,, Discrete Contin. Dyn. Syst. A, 28 (2010), 1669. doi: 10.3934/dcds.2010.28.1669.

[26]

C. Xu and T. Tang, Stability analysis of large time-stepping methods for epitaxial growth models,, SIAM J. Numer. Anal., 44 (2006), 1759. doi: 10.1137/050628143.

[27]

X. Yang, Error analysis of stabilized semi-implicit method of Allen-Cahn equation,, Discrete Contin. Dyn. Syst. Ser. B, 11 (2009), 1057. doi: 10.3934/dcdsb.2009.11.1057.

[28]

J. Zhang and Q. Du, Numerical studies of discrete approximations to the Allen-Cahn equation in the sharp interface limit,, SIAM J. Sci. Comput., 31 (2009), 3042. doi: 10.1137/080738398.

[29]

Z. Zhang and Z. Qiao, An adaptive time-stepping strategy for the Cahn-Hilliard equation,, Commun. Comput. Phys., 11 (2012), 1261. doi: 10.4208/cicp.300810.140411s.

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