August  2014, 19(6): 1737-1747. doi: 10.3934/dcdsb.2014.19.1737

An unconditionally stable numerical method for the viscous Cahn--Hilliard equation

1. 

Institute of Mathematical Sciences, Ewha W. University, Seoul 120-750, South Korea

2. 

Department of Mathematics, Korea University, Seoul 136-713, South Korea, South Korea

Received  November 2013 Revised  March 2014 Published  June 2014

We present an unconditionally stable finite difference method for solving the viscous Cahn--Hilliard equation. We prove the unconditional stability of the proposed scheme by using the decrease of a discrete functional. We present numerical results that validate the convergence and unconditional stability properties of the method. Further, we present numerical experiments that highlight the different temporal evolutions of the Cahn--Hilliard and viscous Cahn--Hilliard equations.
Citation: Jaemin Shin, Yongho Choi, Junseok Kim. An unconditionally stable numerical method for the viscous Cahn--Hilliard equation. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1737-1747. doi: 10.3934/dcdsb.2014.19.1737
References:
[1]

F. Bai, C. M. Elliott, A. Gardiner, A. Spence and A. M. Stuart, The viscous Cahn-Hilliard equation. Part I: Computations,, Nonlinearity, 8 (1995), 131. doi: 10.1088/0951-7715/8/2/002.

[2]

K. Binder, H. L. Frisch and J. Jäckle, Kinetics of phase separation in the presence of slowly relaxing structural variables,, J. Chem. Phys., 85 (1986), 1505. doi: 10.1063/1.451190.

[3]

J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy,, J. Chem. Phys., 28 (1958), 258. doi: 10.1063/1.1744102.

[4]

A. N. Carvalho and T. Dlotko, Dynamics of the viscous Cahn-Hilliard equation,, J. Math. Anal. Appl., 344 (2008), 703. doi: 10.1016/j.jmaa.2008.03.020.

[5]

R. Chella and J. Viñals, Mixing of a two-phase fluid by cavity flow,, Phys. Rev. E., 53 (1996), 3832. doi: 10.1103/PhysRevE.53.3832.

[6]

L. Q. Chen and J. Shen, Applications of semi-implicit Fourier-spectral method to phase field equations,, Comput. Phys. Commun., 108 (1998), 147. doi: 10.1016/S0010-4655(97)00115-X.

[7]

L. Cherfils, A. Miranville and S. Zelik, The Cahn-Hilliard equation with logarithmic potentials,, Milan J. Math., 79 (2011), 561. doi: 10.1007/s00032-011-0165-4.

[8]

S. M. Choo, S. K. Chung and Y. J. Lee, A conservative difference scheme for the viscous Cahn-Hilliard equation with a nonconstant gradient energy coefficient,, Appl. Numer. Math., 51 (2004), 207. doi: 10.1016/j.apnum.2004.02.006.

[9]

P. Colli, G. Gilardi, P. Podio-Guidugli and J. Sprekels, An asymptotic analysis for a nonstandard Cahn-Hilliard system with viscosity,, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 353.

[10]

P. Colli, G. Gilardi, P. Podio-Guidugli and J. Sprekels, Well-posedness and long-time behavior for a nonstandard viscous Cahn-Hilliard system,, SIAM J. Appl. Math., 71 (2011), 1849. doi: 10.1137/110828526.

[11]

S. Dai and Q. Du, Motion of interfaces governed by the Cahn-Hilliard equation with highly disparate diffusion mobility,, SIAM J. Appl. Math., 72 (2012), 1818. doi: 10.1137/120862582.

[12]

Q. Du and R. A. Nicolaides, Numerical analysis of a continuum model of phase transition,, SIAM J. Numer. Anal., 28 (1991), 1310. doi: 10.1137/0728069.

[13]

C. M. Elliott and A. M. Stuart, Viscous Cahn-Hilliard equation II. Analysis,, J. Differential Equations, 128 (1996), 387. doi: 10.1006/jdeq.1996.0101.

[14]

D. J. Eyre, An Unconditionally Stable One-Step Scheme for Gradient Systems,, Unpublished article, (1998).

[15]

D. J. Eyre, Unconditionally gradient stable time marching the Cahn-Hilliard equation,, Mater. Res. Soc. Symp. Proc., 529 (1998), 39. doi: 10.1557/PROC-529-39.

[16]

D. Furihata, A stable and conservative finite difference scheme for the Cahn-Hilliard Equation,, Numer. Math., 87 (2001), 675. doi: 10.1007/PL00005429.

[17]

C. G. Gal and M. Grasselli, Singular limit of viscous Cahn-Hilliard equations with memory and dynamic boundary conditions,, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 1581. doi: 10.3934/dcdsb.2013.18.1581.

[18]

H. Gómez, V. M. Calo, Y. Bazilevs and T. J. Hughes, Isogeometric analysis of the Cahn-Hilliard phase-field model,, Comput. Methods Appl. Mech. Engrg., 197 (2008), 4333.

[19]

M. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance,, Phys. D, 92 (1996), 178. doi: 10.1016/0167-2789(95)00173-5.

[20]

R. A. Horn and C. R. Johnson, Matrix Analysis,, Cambridge Univ. Press, (1985). doi: 10.1017/CBO9780511810817.

[21]

D. Kay and R. Welford, A multigrid finite element solver for the Cahn-Hilliard equation,, J. Comput. Phys., 212 (2006), 288. doi: 10.1016/j.jcp.2005.07.004.

[22]

J. Kim, A continuous surface tension force formulation for diffuse-interface models,, J. Comput. Phys., 204 (2005), 784. doi: 10.1016/j.jcp.2004.10.032.

[23]

J. Kim, A numerical method for the Cahn-Hilliard equation with a variable mobility,, Commun. Nonlinear. Sci. Numer. Simulat., 12 (2007), 1560. doi: 10.1016/j.cnsns.2006.02.010.

[24]

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

[25]

J. S. Kim and H. O. Bae, An unconditionally gradient stable adaptive mesh refinement for the Cahn-Hilliard equation,, J. Korean Phys. Soc., 53 (2008), 672.

[26]

D. Li and X. Ju, On dynamical behavior of viscous Cahn-Hilliard equation,, Discret. Contin. Dyn. Syst., 32 (2013), 2207. doi: 10.3934/dcds.2012.32.2207.

[27]

S. Momani and V. S. Erturk, A numerical scheme for the solution of viscous Cahn-Hilliard equation,, Numer. Meth. Part. D. E., 24 (2008), 663. doi: 10.1002/num.20286.

[28]

A. Novick-Cohen, On the viscous Cahn-Hilliard equation, Material Instabilities in Continuum and Related Mathematical Problems,, Oxford Univ. Press, (1988), 329.

[29]

A. Novick-Cohen, The Cahn-Hillard equation: Mathematical and modeling perspectives,, Adv. Math. Sci. Appl., 8 (1998), 965.

[30]

R. L. Pego, Front migration in the nonlinear Cahn-Hilliard equation,, Proc. R. Soc. Lond. A. Math. Phys. Sci., 422 (1989), 261. doi: 10.1098/rspa.1989.0027.

[31]

M. Pierre, Uniform convergence for a finite-element discretization of a viscous diffusion equation,, J. Numer. Anal., 30 (2010), 487. doi: 10.1093/imanum/drn055.

[32]

L. G. Reyna and M. Ward, Metastable internal layer dynamics for the viscous Cahn-Hilliard equation,, Methods and Appl. of Anal., 2 (1995), 285.

[33]

X. Sun and M. Ward, Dynamics and coarsening of interfaces for the viscous Cahn-Hilliard equation in one spatial dimension,, Stud. Appl. Math., 105 (2000), 203. doi: 10.1111/1467-9590.00149.

[34]

U. Trottenberg, C. Oosterlee and A. Schüller, Multigrid,, Academic press, (2001).

[35]

B. P. Vollmayr-Lee and A. D. Rutenberg, Fast and accurate coarsening simulation with an unconditionally stable time step,, Phys. Rev. E., 68 (2003). doi: 10.1103/PhysRevE.68.066703.

[36]

S. D. Yang, H. G. Lee and J. S. Kim, A phase-field approach for minimizing the area of triply periodic surfaces with volume constraint,, Comput. Phys. Commun., 181 (2010), 1037. doi: 10.1016/j.cpc.2010.02.010.

[37]

P. Yue, C. Zhou and J. J. Feng, Spontaneous shrinkage of drops and mass conservation in phase-field simulations,, J. Comput. Phys., 223 (2007), 1. doi: 10.1016/j.jcp.2006.11.020.

show all references

References:
[1]

F. Bai, C. M. Elliott, A. Gardiner, A. Spence and A. M. Stuart, The viscous Cahn-Hilliard equation. Part I: Computations,, Nonlinearity, 8 (1995), 131. doi: 10.1088/0951-7715/8/2/002.

[2]

K. Binder, H. L. Frisch and J. Jäckle, Kinetics of phase separation in the presence of slowly relaxing structural variables,, J. Chem. Phys., 85 (1986), 1505. doi: 10.1063/1.451190.

[3]

J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy,, J. Chem. Phys., 28 (1958), 258. doi: 10.1063/1.1744102.

[4]

A. N. Carvalho and T. Dlotko, Dynamics of the viscous Cahn-Hilliard equation,, J. Math. Anal. Appl., 344 (2008), 703. doi: 10.1016/j.jmaa.2008.03.020.

[5]

R. Chella and J. Viñals, Mixing of a two-phase fluid by cavity flow,, Phys. Rev. E., 53 (1996), 3832. doi: 10.1103/PhysRevE.53.3832.

[6]

L. Q. Chen and J. Shen, Applications of semi-implicit Fourier-spectral method to phase field equations,, Comput. Phys. Commun., 108 (1998), 147. doi: 10.1016/S0010-4655(97)00115-X.

[7]

L. Cherfils, A. Miranville and S. Zelik, The Cahn-Hilliard equation with logarithmic potentials,, Milan J. Math., 79 (2011), 561. doi: 10.1007/s00032-011-0165-4.

[8]

S. M. Choo, S. K. Chung and Y. J. Lee, A conservative difference scheme for the viscous Cahn-Hilliard equation with a nonconstant gradient energy coefficient,, Appl. Numer. Math., 51 (2004), 207. doi: 10.1016/j.apnum.2004.02.006.

[9]

P. Colli, G. Gilardi, P. Podio-Guidugli and J. Sprekels, An asymptotic analysis for a nonstandard Cahn-Hilliard system with viscosity,, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 353.

[10]

P. Colli, G. Gilardi, P. Podio-Guidugli and J. Sprekels, Well-posedness and long-time behavior for a nonstandard viscous Cahn-Hilliard system,, SIAM J. Appl. Math., 71 (2011), 1849. doi: 10.1137/110828526.

[11]

S. Dai and Q. Du, Motion of interfaces governed by the Cahn-Hilliard equation with highly disparate diffusion mobility,, SIAM J. Appl. Math., 72 (2012), 1818. doi: 10.1137/120862582.

[12]

Q. Du and R. A. Nicolaides, Numerical analysis of a continuum model of phase transition,, SIAM J. Numer. Anal., 28 (1991), 1310. doi: 10.1137/0728069.

[13]

C. M. Elliott and A. M. Stuart, Viscous Cahn-Hilliard equation II. Analysis,, J. Differential Equations, 128 (1996), 387. doi: 10.1006/jdeq.1996.0101.

[14]

D. J. Eyre, An Unconditionally Stable One-Step Scheme for Gradient Systems,, Unpublished article, (1998).

[15]

D. J. Eyre, Unconditionally gradient stable time marching the Cahn-Hilliard equation,, Mater. Res. Soc. Symp. Proc., 529 (1998), 39. doi: 10.1557/PROC-529-39.

[16]

D. Furihata, A stable and conservative finite difference scheme for the Cahn-Hilliard Equation,, Numer. Math., 87 (2001), 675. doi: 10.1007/PL00005429.

[17]

C. G. Gal and M. Grasselli, Singular limit of viscous Cahn-Hilliard equations with memory and dynamic boundary conditions,, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 1581. doi: 10.3934/dcdsb.2013.18.1581.

[18]

H. Gómez, V. M. Calo, Y. Bazilevs and T. J. Hughes, Isogeometric analysis of the Cahn-Hilliard phase-field model,, Comput. Methods Appl. Mech. Engrg., 197 (2008), 4333.

[19]

M. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance,, Phys. D, 92 (1996), 178. doi: 10.1016/0167-2789(95)00173-5.

[20]

R. A. Horn and C. R. Johnson, Matrix Analysis,, Cambridge Univ. Press, (1985). doi: 10.1017/CBO9780511810817.

[21]

D. Kay and R. Welford, A multigrid finite element solver for the Cahn-Hilliard equation,, J. Comput. Phys., 212 (2006), 288. doi: 10.1016/j.jcp.2005.07.004.

[22]

J. Kim, A continuous surface tension force formulation for diffuse-interface models,, J. Comput. Phys., 204 (2005), 784. doi: 10.1016/j.jcp.2004.10.032.

[23]

J. Kim, A numerical method for the Cahn-Hilliard equation with a variable mobility,, Commun. Nonlinear. Sci. Numer. Simulat., 12 (2007), 1560. doi: 10.1016/j.cnsns.2006.02.010.

[24]

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

[25]

J. S. Kim and H. O. Bae, An unconditionally gradient stable adaptive mesh refinement for the Cahn-Hilliard equation,, J. Korean Phys. Soc., 53 (2008), 672.

[26]

D. Li and X. Ju, On dynamical behavior of viscous Cahn-Hilliard equation,, Discret. Contin. Dyn. Syst., 32 (2013), 2207. doi: 10.3934/dcds.2012.32.2207.

[27]

S. Momani and V. S. Erturk, A numerical scheme for the solution of viscous Cahn-Hilliard equation,, Numer. Meth. Part. D. E., 24 (2008), 663. doi: 10.1002/num.20286.

[28]

A. Novick-Cohen, On the viscous Cahn-Hilliard equation, Material Instabilities in Continuum and Related Mathematical Problems,, Oxford Univ. Press, (1988), 329.

[29]

A. Novick-Cohen, The Cahn-Hillard equation: Mathematical and modeling perspectives,, Adv. Math. Sci. Appl., 8 (1998), 965.

[30]

R. L. Pego, Front migration in the nonlinear Cahn-Hilliard equation,, Proc. R. Soc. Lond. A. Math. Phys. Sci., 422 (1989), 261. doi: 10.1098/rspa.1989.0027.

[31]

M. Pierre, Uniform convergence for a finite-element discretization of a viscous diffusion equation,, J. Numer. Anal., 30 (2010), 487. doi: 10.1093/imanum/drn055.

[32]

L. G. Reyna and M. Ward, Metastable internal layer dynamics for the viscous Cahn-Hilliard equation,, Methods and Appl. of Anal., 2 (1995), 285.

[33]

X. Sun and M. Ward, Dynamics and coarsening of interfaces for the viscous Cahn-Hilliard equation in one spatial dimension,, Stud. Appl. Math., 105 (2000), 203. doi: 10.1111/1467-9590.00149.

[34]

U. Trottenberg, C. Oosterlee and A. Schüller, Multigrid,, Academic press, (2001).

[35]

B. P. Vollmayr-Lee and A. D. Rutenberg, Fast and accurate coarsening simulation with an unconditionally stable time step,, Phys. Rev. E., 68 (2003). doi: 10.1103/PhysRevE.68.066703.

[36]

S. D. Yang, H. G. Lee and J. S. Kim, A phase-field approach for minimizing the area of triply periodic surfaces with volume constraint,, Comput. Phys. Commun., 181 (2010), 1037. doi: 10.1016/j.cpc.2010.02.010.

[37]

P. Yue, C. Zhou and J. J. Feng, Spontaneous shrinkage of drops and mass conservation in phase-field simulations,, J. Comput. Phys., 223 (2007), 1. doi: 10.1016/j.jcp.2006.11.020.

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