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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[14]

D. J. Eyre, An Unconditionally Stable One-Step Scheme for Gradient Systems,, Unpublished article, (1998).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

[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. Google Scholar

[16]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[20]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[24]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[28]

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

[29]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[34]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[14]

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

[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. Google Scholar

[16]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[20]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[24]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[28]

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

[29]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[34]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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