Email this article 
A Conservative Numerical Method for Solving the Cahn-Hilliard Equation
- Authors: Galeeva D.R.1, Kireev V.N.1, Kovaleva L.A.1, Musin A.A.1
-
Affiliations:
- Ufa University of Science and Technology
- Issue: Vol 89, No 1 (2025)
- Pages: 136-148
- Section: Articles
- URL: https://ogarev-online.ru/0032-8235/article/view/303583
- DOI: https://doi.org/10.31857/S0032823525010101
- EDN: https://elibrary.ru/BNYENE
- ID: 303583
Cite item
Open Access
Access granted
Subscription Access
Abstract
This paper presents a conservative numerical algorithm for solving the Cahn-Hillard equation. A method for linearizing the Cahn-Hillard equation is proposed, and a numerical scheme is constructed based on the control volume method. The implementation of the proposed numerical algorithm is described in detail. The conservativeness of the proposed discrete scheme is verified by numerical simulation. Numerical experiments were carried out.
About the authors
D. R. Galeeva
Ufa University of Science and Technology
Author for correspondence.
Email: lara_wood@mail.ru
Russian Federation, Ufa
V. N. Kireev
Ufa University of Science and Technology
Email: lara_wood@mail.ru
Russian Federation, Ufa
L. A. Kovaleva
Ufa University of Science and Technology
Email: lara_wood@mail.ru
Russian Federation, Ufa
A. A. Musin
Ufa University of Science and Technology
Email: lara_wood@mail.ru
Russian Federation, Ufa
References
- Nigmatulin R. I. Fundamentals of Mechanics of Heterogeneous Media. Moscow: Nauka, 1978. 336 p.
- Gueyffier D., Li J., Nadim A. et al. Volume‑of‑fluid interface tracking with smoothed surface stress methods for three‑dimensional flows // J. Comput. Phys., 1999, vol. 152, pp. 423–456.
- Glimm J., Grove J. W., Li X. L. et al. Three‑dimensional front tracking // SIAM J. Sci. Comput., 1998, vol. 19, pp. 703–727.
- Anderson D. M., McFadden G. B., Wheeler A. A. Diffuse‑interface methods in fluid mechanics // Annu. Rev. Fluid Mech., 1998, vol. 30, pp. 139–165.
- Du Q., Feng X.‑B. The phase field method for geometric moving interfaces and their numerical approximations // In: Handbook of Numerical Analysis. Vol. 21. Elsevier, 2020, pp. 425–508.
- Badalassi V. E., Ceniceros H. D., Banerjee S. Computation of multiphase systems with phase field models // J. Comput. Phys., 2003, vol. 190, pp. 371–397.
- Cahn J. W., Hilliard J. E. Free energy of a nonuniform system. I. Interfacial free energy // J. Chem. Phys., 1958, vol. 28(2), pp. 258–267.
- Miranville A. The Cahn‑Hilliard equation: recent advances and applications // SIAM, 2019.
- Lovrić A., Dettmer W., Perić D. Low order finite element methods for the Navier‑Stokes–Cahn‑Hilliard equations // arXiv preprint, 2019. arXiv:1911.06718.
- Choo S. M., Chung S. K. Conservative nonlinear difference scheme for the Cahn‑Hilliard equation // Comput. Math. Appl., 1998, vol. 36, pp. 31–39.
- Choo S. M., Chung S. K., Kim K. I. Conservative nonlinear difference scheme for the Cahn‑Hilliard equation. II // Comput. Math. Appl., 2000, vol. 39, pp. 229–243.
- Elliott C. M. The Cahn‑Hilliard model for the kinetics of phase separation // In: Math. Models for Phase Change Problems, 1989, pp. 35–73.
- Eyre D. J. An unconditionally stable one‑step scheme for gradient systems // MRS Proc., 1998.
- Patankar S. V. Numerical Methods for Solving Problems of Heat Transfer and Fluid Dynamics. Moscow: MPEI, 1984. 145 p.
- Dhaouadi F., Dumbser M., Gavrilyuk S. A first‑order hyperbolic reformulation of the Cahn‑Hilliard equation // arXiv preprint, 2024. arXiv:2408.03862.
- Li Y., Jeong D., Shin J., Kim J. A conservative numerical method for the Cahn‑Hilliard equation with Dirichlet boundary conditions in complex domains // Comput. Math. Appl., 2013, vol. 65(1), pp. 102–115.
- Lifshitz I. M., Slezov V. V. Kinetics of precipitation from supersaturated solid solutions // J. Phys. Chem. Solids, 1961, vol. 19(1–2), pp. 35–50.
- Naraigh O. L., Gloster A. A large‑scale statistical study of the coarsening rate in models of Ostwald‑Ripening // arXiv preprint, 2019. arXiv:1911.03386.
Supplementary files
