A practically stable explicit numerical method for the ternary Cahn–Hilliard system

Seokjun Ham; Hyundong Kim; Youngjin Hwang; Junseok Kim; Seokjun Ham; Hyundong Kim; Youngjin Hwang; Junseok Kim

doi:10.3934/mmc.2025019

Mathematical Modelling and Control

2025, Volume 5, Issue 3: 280-291. doi: 10.3934/mmc.2025019

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Research article

A practically stable explicit numerical method for the ternary Cahn–Hilliard system

1.
Department of Mathematics, Korea University, Seoul 02841, Republic of Korea
2.
Department of Mathematics and Physics, Gangneung-Wonju National University, Gangneung 25457, Republic of Korea
3.
Institute for Smart Infrastructure, Gangneung-Wonju National University, Gangneung 25457, Republic of Korea

Received: 16 October 2024 Revised: 06 March 2025 Accepted: 10 March 2025 Published: 24 July 2025

We propose a practically stable explicit finite difference method (FDM) for the ternary Cahn–Hilliard (CH) system, which involves challenging fourth-order nonlinear terms. Therefore, if we use a fully explicit numerical method such as the explicit Euler scheme, then the excessively restrictive time-step limitation makes it impractical to use. The proposed algorithm is based on the alternating direction explicit (ADE) scheme, which ensures both simplicity of implementation and stability of the computational solutions for the ternary CH system. To show the high performance of the proposed algorithm, we present multiple numerical experiments. The numerical tests demonstrate that the proposed method overcomes the stringent time-step limitations inherent in the explicit Euler method.
- ternary Cahn–Hilliard system,
- finite difference method,
- stable explicit numerical method,
- alternating directional explicit scheme
Citation: Seokjun Ham, Hyundong Kim, Youngjin Hwang, Junseok Kim. A practically stable explicit numerical method for the ternary Cahn–Hilliard system[J]. Mathematical Modelling and Control, 2025, 5(3): 280-291. doi: 10.3934/mmc.2025019

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Abstract

We propose a practically stable explicit finite difference method (FDM) for the ternary Cahn–Hilliard (CH) system, which involves challenging fourth-order nonlinear terms. Therefore, if we use a fully explicit numerical method such as the explicit Euler scheme, then the excessively restrictive time-step limitation makes it impractical to use. The proposed algorithm is based on the alternating direction explicit (ADE) scheme, which ensures both simplicity of implementation and stability of the computational solutions for the ternary CH system. To show the high performance of the proposed algorithm, we present multiple numerical experiments. The numerical tests demonstrate that the proposed method overcomes the stringent time-step limitations inherent in the explicit Euler method.

References

[1]	J. E. Morral, J. W. Cahn, Spinodal decomposition in ternary systems, Acta Metall. Sinica, 19 (1971), 1037–1045. https://doi.org/10.1016/0001-6160(71)90036-8 doi: 10.1016/0001-6160(71)90036-8
[2]	S. Bhattacharyya, T. A. Abinandanan, A study of phase separation in ternary alloys, Bull. Mater. Sci., 26 (2003), 193–197. https://doi.org/10.1007/BF02712812 doi: 10.1007/BF02712812
[3]	S. Sugathan, S. Bhattacharya, A phase-field study of elastic stress effects on phase separation in ternary alloys, Comput. Mater. Sci., 172 (2020), 109284. https://doi.org/10.1016/j.commatsci.2019.109284 doi: 10.1016/j.commatsci.2019.109284
[4]	G. I. T$\acute{o}$th, M. Zarifi, B. Kvamme, Phase-field theory of multicomponent incompressible Cahn–Hilliard liquids, Phys. Rev. E, 93 (2016), 013126. https://doi.org/10.1103/PhysRevE.93.013126 doi: 10.1103/PhysRevE.93.013126
[5]	J. M. Park, Mathematical modeling and computational simulation of phase separation in ternary mixtures, Appl. Math. Comput., 330 (2018), 11–22. https://doi.org/10.1016/j.amc.2018.02.006 doi: 10.1016/j.amc.2018.02.006
[6]	H. G. Lee, J. Shin, J. Y. Lee, A high-order convex splitting method for a non-additive Cahn–Hilliard energy functional, Mathematics, 7 (2019), 1242. https://doi.org/10.3390/math7121242 doi: 10.3390/math7121242
[7]	W. Chen, C. Wang, S. Wang, X. Wang, S. M. Wise, Energy stable numerical schemes for ternary Cahn–Hilliard system, J. Sci. Comput., 84 (2020), 1–36. https://doi.org/10.1007/s10915-020-01276-z doi: 10.1007/s10915-020-01276-z
[8]	C. Chen, X. Li, J. Zhang, X. Yang, Efficient linear, decoupled, and unconditionally stable scheme for a ternary Cahn–Hilliard type Nakazawa-Ohta phase-field model for tri-block copolymers, Appl. Math. Comput., 388 (2021), 125463. https://doi.org/10.1016/j.amc.2020.125463 doi: 10.1016/j.amc.2020.125463
[9]	L. Dong, C. Wang, S. M. Wise, Z. Zhang, A positivity-preserving, energy stable scheme for a ternary Cahn–Hilliard system with the singular interfacial parameters, J. Comput. Phys., 442 (2021), 110451. https://doi.org/10.1016/j.jcp.2021.110451 doi: 10.1016/j.jcp.2021.110451
[10]	Z. Wang, J. Zhang, X. Yang, Decoupled, second-order accurate in time and unconditionally energy stable scheme for a hydrodynamically coupled ternary Cahn–Hilliard phase-field model of triblock copolymer melts, Comput. Math. Appl., 124 (2022), 241–257. https://doi.org/10.1016/j.camwa.2022.08.046 doi: 10.1016/j.camwa.2022.08.046
[11]	C. Rohde, L. von Wolff, A ternary Cahn-Hilliard-Navier-Stokes model for two-phase flow with precipitation and dissolution, Math. Mod. Methods Appl. Sci., 31 (2021), 1–35. https://doi.org/10.1142/S0218202521500019 doi: 10.1142/S0218202521500019
[12]	C. Zhan, Z. Chai, B. Shi, A ternary phase-field model for two-phase flows in complex geometries, Phys. D, 460 (2024), 134087. https://doi.org/10.1016/j.physd.2024.134087 doi: 10.1016/j.physd.2024.134087
[13]	J. Manzanero, C. Redondo, G. Rubio, E. Ferrer, $\acute{A}$. Rivero–Jim$\acute{e}$nez, A discontinuous Galerkin approximation for a wall-bounded consistent three-component Cahn–Hilliard flow model, Comput. Fluids, 225 (2021), 104971. https://doi.org/10.1016/j.compfluid.2021.104971 doi: 10.1016/j.compfluid.2021.104971
[14]	M. Rudziva, O. A. Noreldin, P. Sibanda, S. P. Goqo, A bifurcation analysis of multicomponent convection in a rotating low prandtl number fluid with internal heating, Appl. Comput. Math., 21 (2022), 78–100. https://doi.org/10.30546/1683-6154.21.1.2022.78 doi: 10.30546/1683-6154.21.1.2022.78
[15]	C. G. Gal, M. Grasselli, A. Poiatti, J. L. Shomberg, Multi–component Cahn–Hilliard systems with singular potentials: theoretical results, Appl. Math. Optim., 88 (2023), 73. https://doi.org/10.1007/s00245-023-10048-8 doi: 10.1007/s00245-023-10048-8
[16]	S. Zhou, Y. M. Xie, Numerical simulation of three-dimensional multicomponent Cahn–Hilliard systems, Int. J. Mech. Sci., 198 (2021), 106349. https://doi.org/10.1016/j.ijmecsci.2021.106349 doi: 10.1016/j.ijmecsci.2021.106349
[17]	Y. Li, R. Liu, Q. Xia, C. He, Z. Li, First-and second-order unconditionally stable direct discretization methods for multi-component Cahn–Hilliard system on surfaces, J. Comput. Appl. Math., 401 (2022), 113778. https://doi.org/10.1016/j.cam.2021.113778 doi: 10.1016/j.cam.2021.113778
[18]	V. K. Saul'yev, Integration of equations of parabolic type by the method of nets, New York: Pergamon Press, 1964. https://doi.org/10.1016/C2013-0-01754-9
[19]	V. K. Saul'yev, A method of numerical integration of diffusion equations, Dokl. Akad. Nauk, 115 (1957), 1077–1079.
[20]	L. J. Campbell, B. Yin, On the stability of alternating direction explicit methods for advection diffusion equations, Numer. Methods Part. Differ. Equations, 23 (2007), 1429–1444. https://doi.org/10.1002/num.20233 doi: 10.1002/num.20233
[21]	X. Yang, W. Zhao, W. Zhao, Optimal error estimates of a discontinuous Galerkin method for stochastic Allen–Cahn equation driven by multiplicative noise, Commun. Comput. Phys., 36 (2024), 133–159. https://doi.org/10.4208/cicp.OA-2023-0280 doi: 10.4208/cicp.OA-2023-0280
[22]	K. K. Hasanov, T. M. Gasumov, Minimal energy control for the wave equation with non-classical boundary condition, Appl. Comput. Math., 9 (2010), 47–56.
[23]	A. Ashyralyev, A. Erdogan, S. Tekalan, An investigation on finite difference method for the first order partial differential equation with the nonlocal boundary condition, Appl. Comput. Math., 18 (2019), 247–260.
[24]	M. Shen, B. Q. Li, Bubble rising and interaction in ternary fluid flow: a phase field study, RSC Adv., 13 (2023), 3561–3574. https://doi.org/10.1039/D2RA06144A doi: 10.1039/D2RA06144A
[25]	G. Lee, S. Lee, Study on decoupled projection method for Cahn–Hilliard equation, J. Korean Soc. Ind. Appl. Math., 27 (2023), 272–280. https://doi.org/10.12941/jksiam.2023.27.272 doi: 10.12941/jksiam.2023.27.272
[26]	D. J. Eyre, An unconditionally stable one-step scheme for gradient systems, Unpublished article, 1997.
[27]	J. Yang, Z. Tan, J. Kim, Original variables based energy-stable time-dependent auxiliary variable method for the incompressible Navier–Stokes equation, Comput. Fluids, 240 (2022), 105432. https://doi.org/10.1016/j.compfluid.2022.105432 doi: 10.1016/j.compfluid.2022.105432
[28]	H. Kim, C. Lee, S. Yoon, Y. Choi, J. Kim, A fast shape transformation using a phase-field model, Extreme Mech. Lett., 52 (2022), 101633. https://doi.org/10.1016/j.eml.2022.101633 doi: 10.1016/j.eml.2022.101633
[29]	S. Ham, Y. Li, D. Jeong, C. Lee, S. Kwak, Y. Hwang, et al., An explicit adaptive finite difference method for the Cahn–Hilliard equation, J. Nonlinear Sci., 32 (2022), 80. https://doi.org/10.1007/s00332-022-09844-3 doi: 10.1007/s00332-022-09844-3
[30]	J. Yang, J. Kim, Efficient and structure-preserving time-dependent auxiliary variable method for a conservative Allen–Cahn type surfactant system, Eng. Comput., 38 (2022), 5231–5250. https://doi.org/10.1007/s00366-021-01583-5 doi: 10.1007/s00366-021-01583-5
[31]	S. Lee, S. Yoon, C. Lee, S. Kim, H. Kim, J. Yang, et al., Effective time step analysis for the Allen–Cahn equation with a high-order polynomial free energy, Int. J. Numer. Methods Eng., 123 (2022), 4726–4743. https://doi.org/10.1002/nme.7053 doi: 10.1002/nme.7053
[32]	Y. Hwang, C. Lee, S. Kwak, Y. Choi, S. Ham, S. Kang, et al., Benchmark problems for the numerical schemes of the phase-field equations, Discrete Dyn. Nat. Soc., 2022 (2022), 2751592. https://doi.org/10.1155/2022/2751592 doi: 10.1155/2022/2751592
[33]	C. Lee, S. Yoon, J. Park, H. Kim, Y. Li, D. Jeong, et al., Phase-field computations of anisotropic ice crystal growth on a spherical surface, Comput. Math. Appl., 125 (2022), 25–33. https://doi.org/10.1016/j.camwa.2022.08.035 doi: 10.1016/j.camwa.2022.08.035

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