August  2020, 40(8): 4927-4960. doi: 10.3934/dcds.2020206

A structure-preserving scheme for the Allen–Cahn equation with a dynamic boundary condition

1. 

Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University 1-5 Yamadaoka, Suita, Osaka 565-0871, Japan

2. 

Cybermedia Center, Osaka University, 1-32 Machikaneyama, Toyonaka, Osaka 560-0043, Japan

* Corresponding author: okumura@cas.cmc.osaka-u.ac.jp

Received  October 2019 Revised  January 2020 Published  May 2020

We propose a structure-preserving finite difference scheme for the Allen–Cahn equation with a dynamic boundary condition using the discrete variational derivative method [9]. In this method, how to discretize the energy which characterizes the equation is essential. Modifying the conventional manner and using an appropriate summation-by-parts formula, we can use a central difference operator as an approximation of an outward normal derivative on the boundary condition in the scheme. We show the stability and the existence and uniqueness of the solution for the proposed scheme. Also, we give the error estimate for the scheme. Numerical experiments demonstrate the effectiveness of the proposed scheme. Besides, through numerical experiments, we confirm that the long-time behavior of the solution under a dynamic boundary condition may differ from that under the Neumann boundary condition.

Citation: Makoto Okumura, Daisuke Furihata. A structure-preserving scheme for the Allen–Cahn equation with a dynamic boundary condition. Discrete & Continuous Dynamical Systems - A, 2020, 40 (8) : 4927-4960. doi: 10.3934/dcds.2020206
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), 1085-1095.  doi: 10.1016/0001-6160(79)90196-2.  Google Scholar

[2]

L. Calatroni and P. Colli, Global solution to the Allen-Cahn equation with singular potentials and dynamic boundary conditions, Nonlinear Anal., 79 (2013), 12-27.  doi: 10.1016/j.na.2012.11.010.  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. Dyn. Syst., 27 (2010), 1511-1533.  doi: 10.3934/dcds.2010.27.1511.  Google Scholar

[5]

P. Colli and T. Fukao, The Allen-Cahn equation with dynamic boundary conditions and mass constraints, Math. Methods Appl. Sci., 38 (2015), 3950-3967.  doi: 10.1002/mma.3329.  Google Scholar

[6]

P. Colli and J. Sprekels, Optimal control of an Allen-Cahn equation with singular potentials and dynamic boundary condition, SIAM J. Control Optim., 53 (2015), 213-234.  doi: 10.1137/120902422.  Google Scholar

[7]

X. FengH. SongT. Tang and J. Yang, Nonlinear stability of the implicit-explicit methods for the Allen-Cahn equation, Inverse Probl. Imag., 7 (2013), 679-695.  doi: 10.3934/ipi.2013.7.679.  Google Scholar

[8]

T. FukaoS. Yoshikawa and S. Wada, Structure-preserving finite difference schemes for the Cahn-Hilliard equation with dynamic boundary conditions in the one-dimensional case, Commun. Pure Appl. Anal., 16 (2017), 1915-1938.  doi: 10.3934/cpaa.2017093.  Google Scholar

[9] D. Furihata and T. Matsuo, Discrete Variational Derivative Method: A Structure-Preserving Numerical Method for Partial Differential Equations, Chapman & Hall/CRC Numerical Analysis and Scientific Computing, CRC Press, Boca Raton, FL, 2011.   Google Scholar
[10]

C. G. Gal and M. Grasselli, The non-isothermal Allen-Cahn equation with dynamic boundary conditions, Discrete Contin. Dyn. Syst., 22 (2008), 1009-1040.  doi: 10.3934/dcds.2008.22.1009.  Google Scholar

[11]

T. Ide, Some energy preserving finite element schemes based on the discrete variational derivative method, Appl. Math. Comput., 175 (2006), 277-296.  doi: 10.1016/j.amc.2005.07.031.  Google Scholar

[12]

J. KimS. Lee and Y. Choi, A conservative Allen-Cahn equation with a space-time dependent Lagrange multiplier, Int. J. Eng. Sci., 84 (2014), 11-17.  doi: 10.1016/j.ijengsci.2014.06.004.  Google Scholar

[13]

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

[14]

H. G. Lee, High-order and mass conservative methods for the conservative Allen-Cahn equation, Comput. Math. Appl., 72 (2016), 620-631.  doi: 10.1016/j.camwa.2016.05.011.  Google Scholar

[15]

M. Liero, Passing from bulk to bulk-surface evolution in the Allen-Cahn equation, Nonlinear Differ. Equ. Appl., 20 (2013), 919-942.  doi: 10.1007/s00030-012-0189-7.  Google Scholar

[16]

F. Nabet, Convergence of a finite-volume scheme for the Cahn-Hilliard equation with dynamic boundary conditions, IMA J. Numer. Anal., 36 (2016), 1898-1942.  doi: 10.1093/imanum/drv057.  Google Scholar

[17]

M. Okumura, A stable and structure-preserving scheme for a non-local Allen-Cahn equation, Jpn. J. Ind. Appl. Math., 35 (2018), 1245-1281.  doi: 10.1007/s13160-018-0326-8.  Google Scholar

[18]

J. Shen and X. Yang, Numerical approximations of Allen-Cahn and Cahn-Hilliard equations, Discrete Contin. Dyn. Syst., 28 (2010), 1669-1691.  doi: 10.3934/dcds.2010.28.1669.  Google Scholar

[19]

J. Sprekels and H. Wu, A note on parabolic equation with nonlinear dynamical boundary condition, Nonlinear Anal., 72 (2010), 3028-3048.  doi: 10.1016/j.na.2009.11.043.  Google Scholar

[20]

Z. Weng and Q. Zhuang, Numerical approximation of the conservative Allen-Cahn equation by operator splitting method, Math. Method. Appl. Sci., 40 (2017), 4462-4480.  doi: 10.1002/mma.4317.  Google Scholar

[21]

K. Yano and S. Yoshikawa, Structure-preserving finite difference schemes for a semilinear thermoelastic system with second order time derivative, Jpn. J. Ind. Appl. Math., 35 (2018), 1213-1244.  doi: 10.1007/s13160-018-0332-x.  Google Scholar

[22]

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

[23]

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

[24]

S. Yoshikawa, Remarks on energy methods for structure-preserving finite difference schemes – Small data global existence and unconditional error estimate, Appl. Math. Comput., 341 (2019), 80-92.  doi: 10.1016/j.amc.2018.08.030.  Google Scholar

[25]

S. ZhaiZ. Weng and X. Feng, Investigations on several numerical methods for the non-local Allen-Cahn equation, Int. J. Heat Mass Transfer, 87 (2015), 111-118.  doi: 10.1016/j.ijheatmasstransfer.2015.03.071.  Google Scholar

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), 1085-1095.  doi: 10.1016/0001-6160(79)90196-2.  Google Scholar

[2]

L. Calatroni and P. Colli, Global solution to the Allen-Cahn equation with singular potentials and dynamic boundary conditions, Nonlinear Anal., 79 (2013), 12-27.  doi: 10.1016/j.na.2012.11.010.  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. Dyn. Syst., 27 (2010), 1511-1533.  doi: 10.3934/dcds.2010.27.1511.  Google Scholar

[5]

P. Colli and T. Fukao, The Allen-Cahn equation with dynamic boundary conditions and mass constraints, Math. Methods Appl. Sci., 38 (2015), 3950-3967.  doi: 10.1002/mma.3329.  Google Scholar

[6]

P. Colli and J. Sprekels, Optimal control of an Allen-Cahn equation with singular potentials and dynamic boundary condition, SIAM J. Control Optim., 53 (2015), 213-234.  doi: 10.1137/120902422.  Google Scholar

[7]

X. FengH. SongT. Tang and J. Yang, Nonlinear stability of the implicit-explicit methods for the Allen-Cahn equation, Inverse Probl. Imag., 7 (2013), 679-695.  doi: 10.3934/ipi.2013.7.679.  Google Scholar

[8]

T. FukaoS. Yoshikawa and S. Wada, Structure-preserving finite difference schemes for the Cahn-Hilliard equation with dynamic boundary conditions in the one-dimensional case, Commun. Pure Appl. Anal., 16 (2017), 1915-1938.  doi: 10.3934/cpaa.2017093.  Google Scholar

[9] D. Furihata and T. Matsuo, Discrete Variational Derivative Method: A Structure-Preserving Numerical Method for Partial Differential Equations, Chapman & Hall/CRC Numerical Analysis and Scientific Computing, CRC Press, Boca Raton, FL, 2011.   Google Scholar
[10]

C. G. Gal and M. Grasselli, The non-isothermal Allen-Cahn equation with dynamic boundary conditions, Discrete Contin. Dyn. Syst., 22 (2008), 1009-1040.  doi: 10.3934/dcds.2008.22.1009.  Google Scholar

[11]

T. Ide, Some energy preserving finite element schemes based on the discrete variational derivative method, Appl. Math. Comput., 175 (2006), 277-296.  doi: 10.1016/j.amc.2005.07.031.  Google Scholar

[12]

J. KimS. Lee and Y. Choi, A conservative Allen-Cahn equation with a space-time dependent Lagrange multiplier, Int. J. Eng. Sci., 84 (2014), 11-17.  doi: 10.1016/j.ijengsci.2014.06.004.  Google Scholar

[13]

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

[14]

H. G. Lee, High-order and mass conservative methods for the conservative Allen-Cahn equation, Comput. Math. Appl., 72 (2016), 620-631.  doi: 10.1016/j.camwa.2016.05.011.  Google Scholar

[15]

M. Liero, Passing from bulk to bulk-surface evolution in the Allen-Cahn equation, Nonlinear Differ. Equ. Appl., 20 (2013), 919-942.  doi: 10.1007/s00030-012-0189-7.  Google Scholar

[16]

F. Nabet, Convergence of a finite-volume scheme for the Cahn-Hilliard equation with dynamic boundary conditions, IMA J. Numer. Anal., 36 (2016), 1898-1942.  doi: 10.1093/imanum/drv057.  Google Scholar

[17]

M. Okumura, A stable and structure-preserving scheme for a non-local Allen-Cahn equation, Jpn. J. Ind. Appl. Math., 35 (2018), 1245-1281.  doi: 10.1007/s13160-018-0326-8.  Google Scholar

[18]

J. Shen and X. Yang, Numerical approximations of Allen-Cahn and Cahn-Hilliard equations, Discrete Contin. Dyn. Syst., 28 (2010), 1669-1691.  doi: 10.3934/dcds.2010.28.1669.  Google Scholar

[19]

J. Sprekels and H. Wu, A note on parabolic equation with nonlinear dynamical boundary condition, Nonlinear Anal., 72 (2010), 3028-3048.  doi: 10.1016/j.na.2009.11.043.  Google Scholar

[20]

Z. Weng and Q. Zhuang, Numerical approximation of the conservative Allen-Cahn equation by operator splitting method, Math. Method. Appl. Sci., 40 (2017), 4462-4480.  doi: 10.1002/mma.4317.  Google Scholar

[21]

K. Yano and S. Yoshikawa, Structure-preserving finite difference schemes for a semilinear thermoelastic system with second order time derivative, Jpn. J. Ind. Appl. Math., 35 (2018), 1213-1244.  doi: 10.1007/s13160-018-0332-x.  Google Scholar

[22]

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

[23]

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

[24]

S. Yoshikawa, Remarks on energy methods for structure-preserving finite difference schemes – Small data global existence and unconditional error estimate, Appl. Math. Comput., 341 (2019), 80-92.  doi: 10.1016/j.amc.2018.08.030.  Google Scholar

[25]

S. ZhaiZ. Weng and X. Feng, Investigations on several numerical methods for the non-local Allen-Cahn equation, Int. J. Heat Mass Transfer, 87 (2015), 111-118.  doi: 10.1016/j.ijheatmasstransfer.2015.03.071.  Google Scholar

Figure 1.  Numerical solution of (1) with (71) obtained by our scheme
Figure 2.  Time development of total energy
Figure 3.  Numerical solution of (1) with (71) obtained by our scheme
Figure 4.  Time development of total energy
Figure 5.  Numerical solution of (1) with (100) obtained by our scheme
Figure 6.  Time development of total energy
Figure 7.  Numerical solution of (1) with (100) obtained by our scheme
Figure 8.  Time development of total energy
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