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September  2017, 16(5): 1915-1938. doi: 10.3934/cpaa.2017093

Structure-preserving finite difference schemes for the Cahn-Hilliard equation with dynamic boundary conditions in the one-dimensional case

1. 

Department of Mathematics, Kyoto University of Education, 1 Fujinomori, Fukakusa, Fushimi-ku, Kyoto, 612-8522, Japan

2. 

Division of Mathematical Sciences, Department of Integrated Science and Technology, Faculty of Science and Technology, Oita University, 700 Dannoharu, Oita, 870-1192, Japan

3. 

Department of Engineering for Production and Environment, Graduate School of Science and Engineering, Ehime University, Bunkyo-cho 3, Matsuyama, Ehime, 790-8577, Japan

* Corresponding author: yoshikawa@oita-u.ac.jp

Received  October 2016 Revised  April 2017 Published  May 2017

Fund Project: The authors are supported by JSPS KAKENHI Grant-in-Aid for Scientific Research(C), Grant Numbers 26400164 for TF and 16K05234 for SY

The structure-preserving finite difference schemes for the one dimensional Cahn-Hilliard equation with dynamic boundary conditions are studied. A dynamic boundary condition is a sort of transmission condition that includes the time derivative, namely, it is itself a time evolution equation. The Cahn-Hilliard equation with dynamic boundary conditions is well-treated from various viewpoints. The standard type consists of a dynamic boundary condition for the order parameter, and the Neumann boundary condition for the chemical potential. Recently, Goldstein-Miranville-Schimperna proposed a new type of dynamic boundary condition for the Cahn-Hilliard equation. In this article, numerical schemes for the problem with these two kinds of dynamic boundary conditions are introduced. In addition, a mathematical result on the existence of a solution for the scheme with an error estimate is also obtained for the former boundary condition.

Citation: Takeshi Fukao, Shuji Yoshikawa, Saori Wada. Structure-preserving finite difference schemes for the Cahn-Hilliard equation with dynamic boundary conditions in the one-dimensional case. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1915-1938. doi: 10.3934/cpaa.2017093
References:
[1] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011. Google Scholar
[2]

L. CherfilsS. Gatti and A. Miranville, A variational approach to a Cahn-Hilliard model in a domain with nonpermeable walls, J. Math. Sci. (N.Y.), 189 (2013), 604-636. doi: 10.1007/s10958-013-1211-2. Google Scholar

[3]

L. Cherfils and M. Petcu, A numerical analysis of the Cahn-Hilliard equation with non-permeable walls, Numer. Math., 128 (2014), 517-549. doi: 10.1007/s00211-014-0618-0. Google Scholar

[4]

L. CherfilsM. Petcu and M. Pierre, A numerical analysis of the Cahn-Hilliard equation with dynamic boundary conditions, Discrete Contin. Dynam. Sys., 27 (2010), 1511-1533. doi: 10.3934/dcds.2010.27.1511. Google Scholar

[5]

P. Colli and T. Fukao, Cahn-Hilliard equation with dynamic boundary conditions and mass constraint on the boundary, J. Math. Anal. Appl., 429 (2015), 1190-1213. doi: 10.1016/j.jmaa.2015.04.057. Google Scholar

[6]

P. Colli and T. Fukao, Equation and dynamic boundary condition of Cahn-Hilliard type with singular potentials, Nonlinear Anal., 127 (2015), 413-433. doi: 10.1016/j.na.2015.07.011. Google Scholar

[7]

P. Colli and T. Fukao, Nonlinear diffusion equations as asymptotic limits of Cahn-Hilliard systems, J. Differential Equations, 260 (2016), 6930-6959. doi: 10.1016/j.jde.2016.01.032. Google Scholar

[8]

P. ColliG. Gilardi and J. Sprekels, On the Cahn-Hilliard equation with dynamic boundary conditions and a dominating boundary potential, J. Math. Anal. Appl., 419 (2014), 972-994. doi: 10.1016/j.jmaa.2014.05.008. Google Scholar

[9]

P. ColliG. Gilardi and J. Sprekels, A boundary control problem for the pure Cahn-Hilliard equation with dynamic boundary conditions, Adv. Nonlinear Anal., 4 (2015), 311-325. doi: 10.1515/anona-2015-0035. Google Scholar

[10]

P. ColliG. Gilardi and J. Sprekels, A boundary control problem for the viscous Cahn-Hilliard equation with dynamic boundary conditions, Appl. Math. Optim., 73 (2016), 195-225. doi: 10.1007/s00245-015-9299-z. Google Scholar

[11]

T. Fukao, Convergence of Cahn-Hilliard systems to the Stefan problem with dynamic boundary conditions, Asymptot. Anal., 99 (2016), 1-21. doi: 10.3233/ASY-161373. Google Scholar

[12]

T. Fukao, Cahn-Hilliard approach to some degenerate parabolic equations with dynamic boundary conditions, 282-291 in System Modeling and Optimization, Springer, 2016. doi: 10.1007/978-3-319-55795-3_26. Google Scholar

[13]

D. Furihata, A stable and conservative finite difference scheme for the Cahn-Hilliard equation, Numer. Math., 87 (2001), 675-699. doi: 10.1007/PL00005429. Google Scholar

[14]

D. Furihata and T. Matsuo, Discrete Variational Derivative Method, Numerical Analysis and Scientific Computing series, CRC Press/Taylor & Francis, 2010. Google Scholar

[15]

G. GilardiA. Miranville and G. Schimperna, On the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions, Commun. Pure. Appl. Anal., 8 (2009), 881-912. doi: 10.3934/cpaa.2009.8.881. Google Scholar

[16]

G. GilardiA. Miranville and G. Schimperna, Long-time behavior of the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions, Chin. Ann. Math. Ser. B, 31 (2010), 679-712. doi: 10.1007/s11401-010-0602-7. Google Scholar

[17]

G. R. GoldsteinA. Miranville and G. Schimperna, A Cahn-Hilliard model in a domain with non-permeable walls, Phys. D, 240 (2011), 754-766. doi: 10.1016/j.physd.2010.12.007. Google Scholar

[18]

B. Kovács and C. Lubich, Numerical analysis of parabolic problems with dynamic boundary conditions, IMA J. Numer. Anal., 37 (2017), 1-39. doi: 10.1093/imanum/drw015. Google Scholar

[19]

R. Racke and Songmu Zheng, The Cahn-Hilliard equation with dynamic boundary conditions, Adv. Differential Equations, 8 (2003), 83-110. Google Scholar

[20]

Hao Wu and Songmu Zheng, Convergence to equilibrium for the Cahn-Hilliard equation with dynamic boundary conditions, J. Differential Equations, 204 (2004), 511-531. doi: 10.1016/j.jde.2004.05.004. Google Scholar

[21]

Hao Wu and Songmu Zheng, Asymptotic behavior of a Cahn-Hilliard equation with Wentzell boundary conditions and mass conservation, Discrete Contin. Dyn. Syst. Ser. S, 22 (2008), 1041-1063. doi: 10.3934/dcds.2008.22.1041. Google Scholar

[22]

S. Yoshikawa, Energy method for structure-preserving finite difference schemes and some properties of difference quotient, J. Comput. Appl. Math., 311 (2017), 394-413. doi: 10.1016/j.cam.2016.08.008. Google Scholar

[23]

S. Yoshikawa, An error estimate for structure-preserving finite difference scheme for the Falk model system of shape memory alloys, IMA J. Numer. Anal., 37 (2017), 477-504. doi: 10.1093/imanum/drv072. Google Scholar

show all references

References:
[1] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011. Google Scholar
[2]

L. CherfilsS. Gatti and A. Miranville, A variational approach to a Cahn-Hilliard model in a domain with nonpermeable walls, J. Math. Sci. (N.Y.), 189 (2013), 604-636. doi: 10.1007/s10958-013-1211-2. Google Scholar

[3]

L. Cherfils and M. Petcu, A numerical analysis of the Cahn-Hilliard equation with non-permeable walls, Numer. Math., 128 (2014), 517-549. doi: 10.1007/s00211-014-0618-0. Google Scholar

[4]

L. CherfilsM. Petcu and M. Pierre, A numerical analysis of the Cahn-Hilliard equation with dynamic boundary conditions, Discrete Contin. Dynam. Sys., 27 (2010), 1511-1533. doi: 10.3934/dcds.2010.27.1511. Google Scholar

[5]

P. Colli and T. Fukao, Cahn-Hilliard equation with dynamic boundary conditions and mass constraint on the boundary, J. Math. Anal. Appl., 429 (2015), 1190-1213. doi: 10.1016/j.jmaa.2015.04.057. Google Scholar

[6]

P. Colli and T. Fukao, Equation and dynamic boundary condition of Cahn-Hilliard type with singular potentials, Nonlinear Anal., 127 (2015), 413-433. doi: 10.1016/j.na.2015.07.011. Google Scholar

[7]

P. Colli and T. Fukao, Nonlinear diffusion equations as asymptotic limits of Cahn-Hilliard systems, J. Differential Equations, 260 (2016), 6930-6959. doi: 10.1016/j.jde.2016.01.032. Google Scholar

[8]

P. ColliG. Gilardi and J. Sprekels, On the Cahn-Hilliard equation with dynamic boundary conditions and a dominating boundary potential, J. Math. Anal. Appl., 419 (2014), 972-994. doi: 10.1016/j.jmaa.2014.05.008. Google Scholar

[9]

P. ColliG. Gilardi and J. Sprekels, A boundary control problem for the pure Cahn-Hilliard equation with dynamic boundary conditions, Adv. Nonlinear Anal., 4 (2015), 311-325. doi: 10.1515/anona-2015-0035. Google Scholar

[10]

P. ColliG. Gilardi and J. Sprekels, A boundary control problem for the viscous Cahn-Hilliard equation with dynamic boundary conditions, Appl. Math. Optim., 73 (2016), 195-225. doi: 10.1007/s00245-015-9299-z. Google Scholar

[11]

T. Fukao, Convergence of Cahn-Hilliard systems to the Stefan problem with dynamic boundary conditions, Asymptot. Anal., 99 (2016), 1-21. doi: 10.3233/ASY-161373. Google Scholar

[12]

T. Fukao, Cahn-Hilliard approach to some degenerate parabolic equations with dynamic boundary conditions, 282-291 in System Modeling and Optimization, Springer, 2016. doi: 10.1007/978-3-319-55795-3_26. Google Scholar

[13]

D. Furihata, A stable and conservative finite difference scheme for the Cahn-Hilliard equation, Numer. Math., 87 (2001), 675-699. doi: 10.1007/PL00005429. Google Scholar

[14]

D. Furihata and T. Matsuo, Discrete Variational Derivative Method, Numerical Analysis and Scientific Computing series, CRC Press/Taylor & Francis, 2010. Google Scholar

[15]

G. GilardiA. Miranville and G. Schimperna, On the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions, Commun. Pure. Appl. Anal., 8 (2009), 881-912. doi: 10.3934/cpaa.2009.8.881. Google Scholar

[16]

G. GilardiA. Miranville and G. Schimperna, Long-time behavior of the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions, Chin. Ann. Math. Ser. B, 31 (2010), 679-712. doi: 10.1007/s11401-010-0602-7. Google Scholar

[17]

G. R. GoldsteinA. Miranville and G. Schimperna, A Cahn-Hilliard model in a domain with non-permeable walls, Phys. D, 240 (2011), 754-766. doi: 10.1016/j.physd.2010.12.007. Google Scholar

[18]

B. Kovács and C. Lubich, Numerical analysis of parabolic problems with dynamic boundary conditions, IMA J. Numer. Anal., 37 (2017), 1-39. doi: 10.1093/imanum/drw015. Google Scholar

[19]

R. Racke and Songmu Zheng, The Cahn-Hilliard equation with dynamic boundary conditions, Adv. Differential Equations, 8 (2003), 83-110. Google Scholar

[20]

Hao Wu and Songmu Zheng, Convergence to equilibrium for the Cahn-Hilliard equation with dynamic boundary conditions, J. Differential Equations, 204 (2004), 511-531. doi: 10.1016/j.jde.2004.05.004. Google Scholar

[21]

Hao Wu and Songmu Zheng, Asymptotic behavior of a Cahn-Hilliard equation with Wentzell boundary conditions and mass conservation, Discrete Contin. Dyn. Syst. Ser. S, 22 (2008), 1041-1063. doi: 10.3934/dcds.2008.22.1041. Google Scholar

[22]

S. Yoshikawa, Energy method for structure-preserving finite difference schemes and some properties of difference quotient, J. Comput. Appl. Math., 311 (2017), 394-413. doi: 10.1016/j.cam.2016.08.008. Google Scholar

[23]

S. Yoshikawa, An error estimate for structure-preserving finite difference scheme for the Falk model system of shape memory alloys, IMA J. Numer. Anal., 37 (2017), 477-504. doi: 10.1093/imanum/drv072. Google Scholar

Figure 1.  Relationship between $x$ and $k$
Figure 2.  Discrete quantities $A_{1,d}^{(n)}$ and $M_d({{\mathit{\boldsymbol{U}}}^{\left( n \right)}})$
Figure 3.  Discrete quantities $A_{2,d}^{(n)}$ and $M_{2,d}^{(n)}$
Figure 4.  Case 1 for $\gamma =0.005$
Figure 5.  Case 1 for $\gamma =0.001$
Figure 6.  Case 1 for $\gamma =0.0005$
Figure 7.  Case 2 for $\gamma =0.005$
Figure 8.  Case 2 for $\gamma =0.001$
Figure 9.  Case 2 for $\gamma =0.0005$
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