doi: 10.3934/cpaa.2021181
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A second-order accurate structure-preserving scheme for the Cahn-Hilliard equation with a dynamic boundary condition

1. 

Research Institute for Electronic Science, Hokkaido University, N12W7, Kita-Ward, Sapporo, Hokkaido, 060-0812, Japan

2. 

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

3. 

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

4. 

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

* Corresponding author

Received  February 2021 Revised  September 2021 Early access November 2021

Fund Project: This work was partially supported by JSPS KAKENHI, Grant No. JP20KK0308, JP20K03687, JP20K20883, JP21K03309, JP21K20314, and The Sumitomo Foundation, Grant No. 190367

We propose a structure-preserving finite difference scheme for the Cahn–Hilliard equation with a dynamic boundary condition using the discrete variational derivative method (DVDM) proposed by Furihata and Matsuo [14]. In this approach, it is important and essential how to discretize the energy which characterizes the equation. By modifying the conventional manner and using an appropriate summation-by-parts formula, we can use a standard central difference operator as an approximation of an outward normal derivative on the discrete boundary condition of the scheme. We show that our proposed scheme is second-order accurate in space, although the previous structure-preserving scheme proposed by Fukao–Yoshikawa–Wada [13] is first-order accurate in space. Also, we show the stability, the existence, and the uniqueness of the solution for our proposed scheme. Computation examples demonstrate the effectiveness of our proposed scheme. Especially through computation examples, we confirm that numerical solutions can be stably obtained by our proposed scheme.

Citation: Makoto Okumura, Takeshi Fukao, Daisuke Furihata, Shuji Yoshikawa. A second-order accurate structure-preserving scheme for the Cahn-Hilliard equation with a dynamic boundary condition. Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2021181
References:
[1]

J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267.   Google Scholar

[2]

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

[3]

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

[4]

L. CherfilsA. Miranville and S. Zelik, The Cahn–Hilliard equation with logarithmic potentials, Milan J. Math., 79 (2011), 561-596.  doi: 10.1007/s00032-011-0165-4.  Google Scholar

[5]

R. ChillE. Fašangová and J. Prüss, Convergence to steady states of solutions of the Cahn–Hilliard and Caginalp equations with dynamic boundary conditions, Math. Nachr., 279 (2006), 1448-1462.  doi: 10.1002/mana.200410431.  Google Scholar

[6]

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

[7]

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

[8]

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

[9]

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

[10]

Q. Du and R. A. Nicolaides, Numerical analysis of a continuum model of phase transition, SIAM J. Numer. Anal., 28 (1991), 1310-1322.  doi: 10.1137/0728069.  Google Scholar

[11]

C. M. Elliott, The Cahn–Hilliard model for the kinetics of phase separation, in Mathematical Models for Phase Change Problems (ed. J. F. Rodrigues), International Series of Numerical Mathematics, 88, Birkhäuser, 1989.  Google Scholar

[12] S. M. Fallat and C. R. Johnson, Totally Nonnegative Matrices, Princeton University Press, Princeton, 2011.  doi: 10.1515/9781400839018.  Google Scholar
[13]

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

[14] 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
[15]

C. G. Gal, A Cahn–Hilliard model in bounded domains with permeable walls, Math. Methods Appl. Sci., 29 (2006), 2009-2036.  doi: 10.1002/mma.757.  Google Scholar

[16]

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

[17]

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

[18]

H. IsraelA. Miranville and M. Petcu, Numerical analysis of a Cahn–Hilliard type equation with dynamic boundary conditions, Ricerche Mat., 64 (2015), 25-50.  doi: 10.1007/s11587-014-0187-7.  Google Scholar

[19]

A. Miranville and S. Zelik, Exponential attractors for the Cahn–Hilliard equation with dynamic boundary conditions, Math. Methods Appl. Sci., 28 (2005), 709-735.  doi: 10.1002/mma.590.  Google Scholar

[20]

A. Miranville and S. Zelik, The Cahn–Hilliard equation with singular potentials and dynamic boundary conditions, Discrete Contin. Dyn. Syst., 28 (2010), 275-310.  doi: 10.3934/dcds.2010.28.275.  Google Scholar

[21]

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

[22]

F. Nabet, An error estimate for a finite-volume scheme for the Cahn–Hilliard equation with dynamic boundary conditions, Numer. Math., 149 (2021), 185-226.   Google Scholar

[23]

M. Okumura and D. Furihata, A structure-preserving scheme for the Allen-Cahn equation with a dynamic boundary condition, Discrete Contin. Dyn. Syst., 40 (2020), 4927-4960.  doi: 10.3934/dcds.2020206.  Google Scholar

[24]

M. Okumura, T. Fukao, D. Furihata and S. Yoshikawa, Program codes for "A second-order accurate structure-preserving scheme for the Cahn-Hilliard equation with a dynamic boundary condition", Zenodo, https://doi.org/10.5281/zenodo.5541647. Google Scholar

[25]

J. PrüssR. Racke and S. Zheng, Maximal regularity and asymptotic behavior of solutions for the Cahn–Hilliard equation with dynamic boundary conditions, Ann. Mat. Pura Appl., 185 (2006), 627-648.  doi: 10.1007/s10231-005-0175-3.  Google Scholar

[26]

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

[27]

H. Wu and S. 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

[28]

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

[29]

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

[30]

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

[31]

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

show all references

References:
[1]

J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267.   Google Scholar

[2]

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

[3]

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

[4]

L. CherfilsA. Miranville and S. Zelik, The Cahn–Hilliard equation with logarithmic potentials, Milan J. Math., 79 (2011), 561-596.  doi: 10.1007/s00032-011-0165-4.  Google Scholar

[5]

R. ChillE. Fašangová and J. Prüss, Convergence to steady states of solutions of the Cahn–Hilliard and Caginalp equations with dynamic boundary conditions, Math. Nachr., 279 (2006), 1448-1462.  doi: 10.1002/mana.200410431.  Google Scholar

[6]

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

[7]

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

[8]

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

[9]

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

[10]

Q. Du and R. A. Nicolaides, Numerical analysis of a continuum model of phase transition, SIAM J. Numer. Anal., 28 (1991), 1310-1322.  doi: 10.1137/0728069.  Google Scholar

[11]

C. M. Elliott, The Cahn–Hilliard model for the kinetics of phase separation, in Mathematical Models for Phase Change Problems (ed. J. F. Rodrigues), International Series of Numerical Mathematics, 88, Birkhäuser, 1989.  Google Scholar

[12] S. M. Fallat and C. R. Johnson, Totally Nonnegative Matrices, Princeton University Press, Princeton, 2011.  doi: 10.1515/9781400839018.  Google Scholar
[13]

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

[14] 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
[15]

C. G. Gal, A Cahn–Hilliard model in bounded domains with permeable walls, Math. Methods Appl. Sci., 29 (2006), 2009-2036.  doi: 10.1002/mma.757.  Google Scholar

[16]

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

[17]

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

[18]

H. IsraelA. Miranville and M. Petcu, Numerical analysis of a Cahn–Hilliard type equation with dynamic boundary conditions, Ricerche Mat., 64 (2015), 25-50.  doi: 10.1007/s11587-014-0187-7.  Google Scholar

[19]

A. Miranville and S. Zelik, Exponential attractors for the Cahn–Hilliard equation with dynamic boundary conditions, Math. Methods Appl. Sci., 28 (2005), 709-735.  doi: 10.1002/mma.590.  Google Scholar

[20]

A. Miranville and S. Zelik, The Cahn–Hilliard equation with singular potentials and dynamic boundary conditions, Discrete Contin. Dyn. Syst., 28 (2010), 275-310.  doi: 10.3934/dcds.2010.28.275.  Google Scholar

[21]

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

[22]

F. Nabet, An error estimate for a finite-volume scheme for the Cahn–Hilliard equation with dynamic boundary conditions, Numer. Math., 149 (2021), 185-226.   Google Scholar

[23]

M. Okumura and D. Furihata, A structure-preserving scheme for the Allen-Cahn equation with a dynamic boundary condition, Discrete Contin. Dyn. Syst., 40 (2020), 4927-4960.  doi: 10.3934/dcds.2020206.  Google Scholar

[24]

M. Okumura, T. Fukao, D. Furihata and S. Yoshikawa, Program codes for "A second-order accurate structure-preserving scheme for the Cahn-Hilliard equation with a dynamic boundary condition", Zenodo, https://doi.org/10.5281/zenodo.5541647. Google Scholar

[25]

J. PrüssR. Racke and S. Zheng, Maximal regularity and asymptotic behavior of solutions for the Cahn–Hilliard equation with dynamic boundary conditions, Ann. Mat. Pura Appl., 185 (2006), 627-648.  doi: 10.1007/s10231-005-0175-3.  Google Scholar

[26]

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

[27]

H. Wu and S. 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

[28]

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

[29]

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

[30]

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

[31]

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

Figure 1.  Numerical solution by our scheme with $ \Delta x = 1/2 $
Figure 2.  Numerical solution by Fukao-Yoshikawa-Wada scheme with $ \Delta x = 1/2 $
Figure 3.  Numerical solution by our scheme with $ \Delta x = 1/40 $
Figure 4.  Numerical solution by Fukao-Yoshikawa-Wada scheme with $ \Delta x = 1/40 $
Figure 5.  Time development of $ M_{\rm d}(\boldsymbol{U}^{(n)}) $ obtained by our scheme with $ \Delta x = 1/40 $: $ M_{\rm d}(\boldsymbol{U}^{(n)}) $ is preserved to accuracy $ 10^{-11} $
Figure 6.  Time development of $ E_{\rm d}^{(n)} - J_{\rm d}(\boldsymbol{U}^{(0)}) $ obtained by our scheme with $ \Delta x = 1/40 $: $ E_{\rm d}^{(n)} $ is preserved to accuracy $ 10^{-6} $
Figure 7.  The discrete $ L^{\infty} $-norm error $ \|\boldsymbol{e}_{\Delta x} \|_{L_{\rm d}^{\infty}} $ versus the space mesh size $ \Delta x $ at time $ T = 400 $: our scheme is second-order accurate in space
Figure 8.  The discrete $ L^{\infty} $-norm error $ \|\boldsymbol{e}_{\Delta t} \|_{L_{\rm d}^{\infty}} $ versus the time mesh size $ \Delta t $ at time $ T = 400 $: our scheme is second-order accurate in time
Figure 9.  Numerical solution by our scheme with $ \Delta x = 1/25 $
Figure 10.  Numerical solution by Fukao-Yoshikawa-Wada scheme with $ \Delta x = 1/25 $
Figure 11.  Numerical solution by our scheme with $ \Delta x = 1/50 $
Figure 12.  Numerical solution by Fukao-Yoshikawa-Wada scheme with $ \Delta x = 1/50 $
Figure 13.  Time development of $ {M_{\rm{d}}}({\mathit{\boldsymbol{U}}^{(n)}}) $ obtained by our scheme with $ \Delta x = 1/50$: ${M_{\rm{d}}}({\mathit{\boldsymbol{U}}^{(n)}})$ is preserved to accuracy 10−14
Figure 14.  Time development of $ E_{\rm{d}}^{(n)} - {J_{\rm{d}}}({\mathit{\boldsymbol{U}}^{(0)}}) $ obtained by our scheme with $ \Delta x = 1/50$: $E_{\rm{d}}^{(n)}$ is preserved to accuracy 10−11
Figure 15.  The discrete L-norm error ${\left\| {{\mathit{\boldsymbol{e}}_{\Delta x}}} \right\|_{L_{\rm{d}}^\infty }}$ versus the space mesh size Δx at time T = 1000: our scheme is second-order accurate in space
Figure 16.  The discrete L-norm error ${\left\| {{\mathit{\boldsymbol{e}}_{\Delta t}}} \right\|_{L_{\rm{d}}^\infty }}$ versus the time mesh size Δt at time T = 1000: the convergence rates of our scheme approach three as Δt decreases
Figure 17.  Numerical solution to (1.1)–(1.2) with (1.5) and (6.1) obtained by our scheme
Figure 18.  Time development of ${M_{\rm{d}}}({\mathit{\boldsymbol{U}}^{(n)}})$ obtained by our scheme: ${M_{\rm{d}}}({\mathit{\boldsymbol{U}}^{(n)}})$ is preserved to accuracy 10−11
Figure 19.  Time development of $E_{_{\rm{d}}}^{(n)} - {J_{\rm{d}}}({\mathit{\boldsymbol{U}}^{(0)}})$ obtained by our scheme: $E_{_{\rm{d}}}^{(n)}$ is preserved to accuracy 10−10
Figure 20.  Numerical solution to (1.1)–(1.2) with (7.16) obtained by the discrete variational derivative scheme
Figure 21.  Time development of ${M_{\rm{d}}}({\mathit{\boldsymbol{U}}^{(n)}})$ obtained by the discrete variational derivative scheme: ${M_{\rm{d}}}({\mathit{\boldsymbol{U}}^{(n)}})$ is preserved to accuracy 10−14
Figure 22.  Time development of $A_{_{\rm{d}}}^{(n)} - {{\bar J}_{\rm{d}}}({\mathit{\boldsymbol{U}}^{(0)}})$ obtained by the discrete variational derivative scheme: $A_{_{\rm{d}}}^{(n)}$ is preserved to accuracy 10−9
Table 1.  The discrete $ L^{\infty} $-norm error $ \|\mathit{\boldsymbol{e}}_{\Delta x} \|_{L_{\rm d}^{\infty}} $ and the convergence rates $ \log_{2}(\|\mathit{\boldsymbol{e}}_{2\Delta x} \|_{L_{\rm d}^{\infty}}/\|\mathit{\boldsymbol{e}}_{\Delta x} \|_{L_{\rm d}^{\infty}}) $ at time $ T = 400 $
$ \Delta x $ $ 2^{-1} $ $ 2^{-2} $ $ 2^{-3} $ $ 2^{-4} $
$ \|\mathit{\boldsymbol{e}}_{\Delta x} \|_{L_{\rm d}^{\infty}} $ 3.5272e-3 8.6474e-4 2.1507e-4 5.1156e-5
Rate / 2.0282 2.0075 2.0718
$ \Delta x $ $ 2^{-1} $ $ 2^{-2} $ $ 2^{-3} $ $ 2^{-4} $
$ \|\mathit{\boldsymbol{e}}_{\Delta x} \|_{L_{\rm d}^{\infty}} $ 3.5272e-3 8.6474e-4 2.1507e-4 5.1156e-5
Rate / 2.0282 2.0075 2.0718
Table 2.  The discrete $ L^{\infty} $-norm error $ \|\mathit{\boldsymbol{e}}_{\Delta t} \|_{L_{\rm d}^{\infty}} $ and the convergence rates $ \log_{2}(\|\mathit{\boldsymbol{e}}_{2\Delta t} \|_{L_{\rm d}^{\infty}}/\|\mathit{\boldsymbol{e}}_{\Delta t} \|_{L_{\rm d}^{\infty}}) $ at time $ T = 400 $
$ \Delta t $ $ 2^{-1} $ $ 2^{-2} $ $ 2^{-3} $ $ 2^{-4} $
$ \|\mathit{\boldsymbol{e}}_{\Delta t} \|_{L_{\rm d}^{\infty}} $ 2.2345e-6 5.6404e-7 1.4274e-7 3.4246e-8
Rate / 1.9861 1.9824 2.0594
$ \Delta t $ $ 2^{-1} $ $ 2^{-2} $ $ 2^{-3} $ $ 2^{-4} $
$ \|\mathit{\boldsymbol{e}}_{\Delta t} \|_{L_{\rm d}^{\infty}} $ 2.2345e-6 5.6404e-7 1.4274e-7 3.4246e-8
Rate / 1.9861 1.9824 2.0594
Table 3.  The discrete $ L^{\infty} $-norm error $ \|\mathit{\boldsymbol{e}}_{\Delta x} \|_{L_{\rm d}^{\infty}} $ and the convergence rates $ \log_{2}(\|\mathit{\boldsymbol{e}}_{2\Delta x} \|_{L_{\rm d}^{\infty}}/\|\mathit{\boldsymbol{e}}_{\Delta x} \|_{L_{\rm d}^{\infty}}) $ at time $ T = 1000 $
$ \Delta x $ $ 2^{-2} $ $ 2^{-3} $ $ 2^{-4} $ $ 2^{-5} $
$ \|\mathit{\boldsymbol{e}}_{\Delta x} \|_{L_{\rm d}^{\infty}} $ 1.7727e-3 4.3813e-4 1.0850e-4 2.5856e-5
Rate / 2.0165 2.0137 2.0691
$ \Delta x $ $ 2^{-2} $ $ 2^{-3} $ $ 2^{-4} $ $ 2^{-5} $
$ \|\mathit{\boldsymbol{e}}_{\Delta x} \|_{L_{\rm d}^{\infty}} $ 1.7727e-3 4.3813e-4 1.0850e-4 2.5856e-5
Rate / 2.0165 2.0137 2.0691
Table 4.  The discrete $ L^{\infty} $-norm error $ \|\mathit{\boldsymbol{e}}_{\Delta t} \|_{L_{\rm d}^{\infty}} $ and the convergence rates $ \log_{2}(\|\mathit{\boldsymbol{e}}_{2\Delta t} \|_{L_{\rm d}^{\infty}}/\|\mathit{\boldsymbol{e}}_{\Delta t} \|_{L_{\rm d}^{\infty}}) $ at time $ T = 1000 $
$ \Delta t $ $ 1/10 $ $ 1/20 $ $ 1/40 $ $ 1/80 $
$ \|\mathit{\boldsymbol{e}}_{\Delta t} \|_{L_{\rm d}^{\infty}} $ 1.2473e-3 4.3482e-4 5.1131e-5 5.2106e-6
Rate / 1.5203 3.0881 3.2947
$ \Delta t $ $ 1/10 $ $ 1/20 $ $ 1/40 $ $ 1/80 $
$ \|\mathit{\boldsymbol{e}}_{\Delta t} \|_{L_{\rm d}^{\infty}} $ 1.2473e-3 4.3482e-4 5.1131e-5 5.2106e-6
Rate / 1.5203 3.0881 3.2947
[1]

Makoto Okumura, Daisuke Furihata. A structure-preserving scheme for the Allen–Cahn equation with a dynamic boundary condition. Discrete & Continuous Dynamical Systems, 2020, 40 (8) : 4927-4960. doi: 10.3934/dcds.2020206

[2]

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

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[8]

Laurence Cherfils, Madalina Petcu, Morgan Pierre. A numerical analysis of the Cahn-Hilliard equation with dynamic boundary conditions. Discrete & Continuous Dynamical Systems, 2010, 27 (4) : 1511-1533. doi: 10.3934/dcds.2010.27.1511

[9]

Gianni Gilardi, A. Miranville, Giulio Schimperna. On the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions. Communications on Pure & Applied Analysis, 2009, 8 (3) : 881-912. doi: 10.3934/cpaa.2009.8.881

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