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A second-order accurate structure-preserving scheme for the Cahn-Hilliard equation with a dynamic boundary condition

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    * Corresponding author 

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

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  • 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.

    Mathematics Subject Classification: 65M06, 65M12.

    Citation:

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  • 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
     | Show Table
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    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV
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