# American Institute of Mathematical Sciences

February  2022, 21(2): 355-392. doi: 10.3934/cpaa.2021181

## 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 Published  February 2022 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 and Applied Analysis, 2022, 21 (2) : 355-392. 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. [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. [3] L. Cherfils, M. 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. [4] L. Cherfils, A. 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. [5] R. Chill, E. 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. [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. [7] P. Colli, G. 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. [8] P. Colli, G. 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. [9] P. Colli, G. 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. [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. [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. [12] S. M. Fallat and C. R. Johnson, Totally Nonnegative Matrices, Princeton University Press, Princeton, 2011.  doi: 10.1515/9781400839018. [13] T. Fukao, S. 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. [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. [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. [16] G. Gilardi, A. 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. [17] G. Gilardi, A. 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. [18] H. Israel, A. 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. [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. [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. [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. [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. [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. [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. [25] J. Prüss, R. 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. [26] R. Racke and S. Zheng, The Cahn–Hilliard equation with dynamic boundary conditions, Adv. Differential Equ., 8 (2003), 83-110. [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. [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. [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. [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. [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.

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##### 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. [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. [3] L. Cherfils, M. 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. [4] L. Cherfils, A. 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. [5] R. Chill, E. 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. [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. [7] P. Colli, G. 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. [8] P. Colli, G. 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. [9] P. Colli, G. 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. [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. [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. [12] S. M. Fallat and C. R. Johnson, Totally Nonnegative Matrices, Princeton University Press, Princeton, 2011.  doi: 10.1515/9781400839018. [13] T. Fukao, S. 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. [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. [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. [16] G. Gilardi, A. 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. [17] G. Gilardi, A. 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. [18] H. Israel, A. 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. [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. [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. [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. [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. [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. [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. [25] J. Prüss, R. 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. [26] R. Racke and S. Zheng, The Cahn–Hilliard equation with dynamic boundary conditions, Adv. Differential Equ., 8 (2003), 83-110. [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. [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. [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. [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. [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.
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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