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

i

This article has been updated. A Correction to this article was published on .

i

The Original Article was published on

i

The Original Article was published on

i

This article has been updated

i

This article was retracted on

i

This article has been updated

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.

Keywords

Modica–Mortola model / high-order multi-well free energy / multi-phase system / Fourier spectral approach / stable scheme

CHANGE HISTORY

A Correction to this paper has been published:
https://doi.org

The Retraction Note was updated to reflect reception of statements of disagreement from all authors.

This article has been retracted.Please see the Retraction Notice for more detail:
https://doi.org

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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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