An unconditionally stable hybrid numerical method for the gradient flow for the high-order Modica–Mortola functional

Hyundong Kim; Zhengang Li; Xinpei Wu; Hyunho Shin; Yunjae Nam; Junseok Kim; Hyundong Kim; Zhengang Li; Xinpei Wu; Hyunho Shin; Yunjae Nam; Junseok Kim

doi:10.3934/era.2025281

Electronic Research Archive

2025, Volume 33, Issue 10: 6375-6390. doi: 10.3934/era.2025281

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An unconditionally stable hybrid numerical method for the gradient flow for the high-order Modica–Mortola functional

1.
Department of Mathematics and Physics, Gangneung-Wonju National University, Gangneung 25457, Republic of Korea
2.
Institute for Smart Infrastructure, Gangneung-Wonju National University, Gangneung 25457, Republic of Korea
3.
Program in Actuarial Science and Financial Engineering, Korea University, Seoul 02841, Republic of Korea
4.
Department of Mathematics, Korea University, Seoul 02841, Republic of Korea

Received: 26 August 2025 Revised: 20 October 2025 Accepted: 20 October 2025 Published: 27 October 2025

This paper presents a numerically stable, time-accurate algorithm for simulating the gradient flow associated with the Modica–Mortola functional with a uniformly spaced multi-well potential. The scheme uses operator splitting; the nonlinear component is updated analytically, while the linear part is advanced by a Fourier spectral discretization. The method is unconditionally stable, preserves pointwise boundedness independently of the time step size, and attains spectral accuracy in space and first-order accuracy in time. We provide a theoretical analysis establishing unconditional stability and boundedness, and present comprehensive numerical experiments that demonstrate the accuracy and robustness of the proposed approach.
- Modica–Mortola model,
- high-order multi-well free energy,
- multi-phase system,
- Fourier spectral approach,
- stable scheme
Citation: Hyundong Kim, Zhengang Li, Xinpei Wu, Hyunho Shin, Yunjae Nam, Junseok Kim. An unconditionally stable hybrid numerical method for the gradient flow for the high-order Modica–Mortola functional[J]. Electronic Research Archive, 2025, 33(10): 6375-6390. doi: 10.3934/era.2025281

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Abstract

This paper presents a numerically stable, time-accurate algorithm for simulating the gradient flow associated with the Modica–Mortola functional with a uniformly spaced multi-well potential. The scheme uses operator splitting; the nonlinear component is updated analytically, while the linear part is advanced by a Fourier spectral discretization. The method is unconditionally stable, preserves pointwise boundedness independently of the time step size, and attains spectral accuracy in space and first-order accuracy in time. We provide a theoretical analysis establishing unconditional stability and boundedness, and present comprehensive numerical experiments that demonstrate the accuracy and robustness of the proposed approach.

References

[1]	L. Modica, S. Mortola, Un esempio di Gamma-convergenza, Boll. Unione Mat. Ital. B, 14 (1977), 285–299.
[2]	H. Wilbuer, P. Kurzeja, J. Mosler, Phase field modeling of hyperelastic material interfaces–Theory, implementation and application to phase transformations, Comput. Methods Appl. Mech. Eng., 426 (2024), 116972. https://doi.org/10.1016/j.cma.2024.116972 doi: 10.1016/j.cma.2024.116972
[3]	A. Aspri, A phase-field approach for detecting cavities via a Kohn–Vogelius type functional, Inverse Probl., 38 (2022), 094001. https://doi.org/10.1088/1361-6420/ac82e4 doi: 10.1088/1361-6420/ac82e4
[4]	J. Wang, J. Kim, A novel modified Modica–Mortola equation with a phase-dependent interfacial function, Int. J. Mod. Phys. B, 36 (2022), 2250055. https://doi.org/10.1142/S0217979222500552 doi: 10.1142/S0217979222500552
[5]	Y. Li, S. Yoon, J. Wang, J. Park, S. Kim, C. Lee, et al., Fast and efficient numerical finite difference method for multiphase image segmentation, Math. Probl. Eng., (2021), 2414209. https://doi.org/10.1155/2021/2414209
[6]	Y. Giga, J. Okamoto, M. Uesaka, A finer singular limit of a single-well Modica–Mortola functional and its applications to the Kobayashi–Warren–Carter energy, Adv. Calc. Var., 16 (2023), 163–182. https://doi.org/10.1515/acv-2020-0113 doi: 10.1515/acv-2020-0113
[7]	Y. Jin, S. Kwak, S. Ham, J. Kim, A fast and efficient numerical algorithm for image segmentation and denoising, AIMS Math., 9 (2024), 5015–5027. https://doi.org/10.3934/math.2024243 doi: 10.3934/math.2024243
[8]	Y. M. Jung, S. H. Kang, J. Shen, Multiphase image segmentation via Modica–Mortola phase transition, SIAM J. Appl. Math., 67 (2007), 1213–1232. https://doi.org/10.1137/060662708 doi: 10.1137/060662708
[9]	B. Bogosel, D. Bucur, I. Fragalà, Phase field approach to optimal packing problems and related Cheeger clusters, Appl. Math. Optim., 81 (2020), 63–87. https://doi.org/10.1007/s00245-018-9476-y doi: 10.1007/s00245-018-9476-y
[10]	S. Ham, S. Kwak, C. Lee, G. Lee, J. Kim, A second-order time-accurate unconditionally stable method for a gradient flow for the Modica–Mortola functional, J. Sci. Comput., 95 (2023), 63. https://doi.org/10.1007/s10915-023-02198-2 doi: 10.1007/s10915-023-02198-2
[11]	Z. Cao, Z. Weng, S. Zhai, An effective operator splitting scheme for general motion by mean curvature using a modified Allen–Cahn equation, Appl. Math. Lett., 163 (2025), 109472. https://doi.org/10.1016/j.aml.2025.109472 doi: 10.1016/j.aml.2025.109472
[12]	G. Peng, Y. Li, An energy stable bound-preserving finite volume scheme for the Allen-Cahn equation based on operator splitting method, Comput. Math. Appl., 178 (2025), 47–60. https://doi.org/10.1016/j.camwa.2024.11.014 doi: 10.1016/j.camwa.2024.11.014
[13]	J. Kim, Modified Wave-Front Propagation and Dynamics Coming from Higher-Order Double-Well Potentials in the Allen–Cahn Equations, Mathematics, 12 (2024), 3796. https://doi.org/10.3390/math12233796 doi: 10.3390/math12233796
[14]	Y. Hwang, I. Kim, S. Kwak, S. Ham, S. Kim, J. Kim, Unconditionally stable monte carlo simulation for solving the multi-dimensional Allen–Cahn equation, Electron. Res. Arch., 31 (2023), 5104–5123. https://doi.org/10.3934/era.2023261 doi: 10.3934/era.2023261
[15]	Y. Hwang, S. Ham, C. Lee, G. Lee, S. Kang, J. Kim, A simple and efficient numerical method for the Allen–Cahn equation on effective symmetric triangular meshes, Electron. Res. Arch., 31 (2023), 4557–4578. https://doi.org/10.3934/era.2023233 doi: 10.3934/era.2023233
[16]	H. G. Lee, A semi-analytical Fourier spectral method for the Swift–Hohenberg equation, Comput. Math. Appl., 74 (2017), 1885–1896. https://doi.org/10.1016/j.camwa.2017.06.053 doi: 10.1016/j.camwa.2017.06.053
[17]	J. Choi, S. Ham, S. Kwak, Y. Hwang, J. Kim, Stability analysis of an explicit numerical scheme for the Allen–Cahn equation with high-order polynomial potentials, AIMS Math., 9 (2024), 19332–19344. https://doi.org/10.3934/math.2024941 doi: 10.3934/math.2024941
[18]	C. Zhang, H. Wang, J. Huang, C. Wang, X. Yue, A second order operator splitting numerical scheme for the "good" Boussinesq equation, Appl. Numer. Math., 119 (2017), 179–193. https://doi.org/10.1016/j.apnum.2017.04.006 doi: 10.1016/j.apnum.2017.04.006
[19]	D. Li, C. Quan, J. Xu, Stability and convergence of Strang splitting. Part I: scalar Allen–Cahn equation, J. Comput. Phys., 458 (2022), 111087. https://doi.org/10.1016/j.jcp.2022.111087 doi: 10.1016/j.jcp.2022.111087

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