# American Institute of Mathematical Sciences

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:

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##### References:
Numerical solution by our scheme with $\Delta x = 1/2$
Numerical solution by Fukao-Yoshikawa-Wada scheme with $\Delta x = 1/2$
Numerical solution by our scheme with $\Delta x = 1/40$
Numerical solution by Fukao-Yoshikawa-Wada scheme with $\Delta x = 1/40$
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 $10^{-11}$
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 $10^{-6}$
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
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
Numerical solution by our scheme with $\Delta x = 1/25$
Numerical solution by Fukao-Yoshikawa-Wada scheme with $\Delta x = 1/25$
Numerical solution by our scheme with $\Delta x = 1/50$
Numerical solution by Fukao-Yoshikawa-Wada scheme with $\Delta x = 1/50$
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
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
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
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
Numerical solution to (1.1)–(1.2) with (1.5) and (6.1) obtained by our scheme
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
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
Numerical solution to (1.1)–(1.2) with (7.16) obtained by the discrete variational derivative scheme
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
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
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
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
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
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
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