Stability and convergence analysis for the implicit-explicit method to the Cahn-Hilliard equation

Li, Dong; Quan, Chaoyu; Tang, Tao

doi:10.1090/mcom/3704

Stability and convergence analysis for the implicit-explicit method to the Cahn-Hilliard equation
HTML articles powered by AMS MathViewer

by Dong Li, Chaoyu Quan and Tao Tang HTML | PDF

Math. Comp. 91 (2022), 785-809 Request permission

Abstract:

Implicit-explicit methods have been successfully used for the efficient numerical simulation of phase field problems such as the Cahn-Hilliard equation or thin film type equations. Due to the lack of maximum principle and stiffness caused by the effect of small dissipation coefficient, most existing theoretical analysis relies on adding additional stabilization terms, mollifying the nonlinearity or introducing auxiliary variables which implicitly either changes the structure of the problem or trades accuracy for stability in a subtle way. In this work, we introduce a robust theoretical framework to analyze directly the stability and accuracy of the standard implicit-explicit approach without stabilization or any other modification. We take the Cahn-Hilliard equation as a model case and provide a rigorous stability and convergence analysis for the original semi-discrete scheme under certain time step constraints. These settle several questions which have been open since the work of Chen and Shen [Comput. Phys. Comm. 108 (1998), pp. 147–158].

References

A. Baskaran, J. S. Lowengrub, C. Wang, and S. M. Wise, Convergence analysis of a second order convex splitting scheme for the modified phase field crystal equation, SIAM J. Numer. Anal. 51 (2013), no. 5, 2851–2873. MR 3118257, DOI 10.1137/120880677
J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial energy free energy, J. Chem. Phys. 28 (1958) 258–267.
L. Q. Chen and J. Shen, Applications of semi-implicit Fourier-spectral method to phase field equations, Comput. Phys. Commun. 108 (1998), 147–158.
Andrew Christlieb, Jaylan Jones, Keith Promislow, Brian Wetton, and Mark Willoughby, High accuracy solutions to energy gradient flows from material science models. part A, J. Comput. Phys. 257 (2014), no. part A, 193–215. MR 3129531, DOI 10.1016/j.jcp.2013.09.049
C. M. Elliott, The Cahn-Hilliard model for the kinetics of phase separation, Mathematical models for phase change problems (Óbidos, 1988) Internat. Ser. Numer. Math., vol. 88, Birkhäuser, Basel, 1989, pp. 35–73. MR 1038064
C. M. Elliott and A. M. Stuart, The global dynamics of discrete semilinear parabolic equations, SIAM J. Numer. Anal. 30 (1993), no. 6, 1622–1663. MR 1249036, DOI 10.1137/0730084
David J. Eyre, Unconditionally gradient stable time marching the Cahn-Hilliard equation, Computational and mathematical models of microstructural evolution (San Francisco, CA, 1998) Mater. Res. Soc. Sympos. Proc., vol. 529, MRS, Warrendale, PA, 1998, pp. 39–46. MR 1676409, DOI 10.1557/PROC-529-39
Xinlong Feng, Tao Tang, and Jiang Yang, Long time numerical simulations for phase-field problems using $p$-adaptive spectral deferred correction methods, SIAM J. Sci. Comput. 37 (2015), no. 1, A271–A294. MR 3304267, DOI 10.1137/130928662
Hector Gomez and Thomas J. R. Hughes, Provably unconditionally stable, second-order time-accurate, mixed variational methods for phase-field models, J. Comput. Phys. 230 (2011), no. 13, 5310–5327. MR 2799512, DOI 10.1016/j.jcp.2011.03.033
Zhen Guan, Cheng Wang, and Steven M. Wise, A convergent convex splitting scheme for the periodic nonlocal Cahn-Hilliard equation, Numer. Math. 128 (2014), no. 2, 377–406. MR 3259494, DOI 10.1007/s00211-014-0608-2
Jing Guo, Cheng Wang, Steven M. Wise, and Xingye Yue, An $H^2$ convergence of a second-order convex-splitting, finite difference scheme for the three-dimensional Cahn-Hilliard equation, Commun. Math. Sci. 14 (2016), no. 2, 489–515. MR 3436249, DOI 10.4310/CMS.2016.v14.n2.a8
Yinnian He, Yunxian Liu, and Tao Tang, On large time-stepping methods for the Cahn-Hilliard equation, Appl. Numer. Math. 57 (2007), no. 5-7, 616–628. MR 2322435, DOI 10.1016/j.apnum.2006.07.026
Fei Liu and Jie Shen, Stabilized semi-implicit spectral deferred correction methods for Allen-Cahn and Cahn-Hilliard equations, Math. Methods Appl. Sci. 38 (2015), no. 18, 4564–4575. MR 3449617, DOI 10.1002/mma.2869
Dong Li, Zhonghua Qiao, and Tao Tang, Characterizing the stabilization size for semi-implicit Fourier-spectral method to phase field equations, SIAM J. Numer. Anal. 54 (2016), no. 3, 1653–1681. MR 3507555, DOI 10.1137/140993193
Dong Li, Zhonghua Qiao, and Tao Tang, Gradient bounds for a thin film epitaxy equation, J. Differential Equations 262 (2017), no. 3, 1720–1746. MR 3582210, DOI 10.1016/j.jde.2016.10.025
Dong Li and Zhonghua Qiao, On second order semi-implicit Fourier spectral methods for 2D Cahn-Hilliard equations, J. Sci. Comput. 70 (2017), no. 1, 301–341. MR 3592143, DOI 10.1007/s10915-016-0251-4
Dong Li and Zhonghua Qiao, On the stabilization size of semi-implicit Fourier-spectral methods for 3D Cahn-Hilliard equations, Commun. Math. Sci. 15 (2017), no. 6, 1489–1506. MR 3668944, DOI 10.4310/CMS.2017.v15.n6.a1
Dong Li and Tao Tang, Stability of the semi-implicit method for the Cahn-Hilliard equation with logarithmic potentials, Ann. Appl. Math. 37 (2021), no. 1, 31–60. MR 4284064, DOI 10.4208/aam.OA-2020-0003
Dong Li, Effective maximum principles for spectral methods, Ann. Appl. Math. 37 (2021), no. 2, 131–290. MR 4294331, DOI 10.4208/aam.oa-2021-0003
Carola-Bibiane Schönlieb and Andrea Bertozzi, Unconditionally stable schemes for higher order inpainting, Commun. Math. Sci. 9 (2011), no. 2, 413–457. MR 2815678, DOI 10.4310/CMS.2011.v9.n2.a4
Jie Shen and Xiaofeng Yang, Numerical approximations of Allen-Cahn and Cahn-Hilliard equations, Discrete Contin. Dyn. Syst. 28 (2010), no. 4, 1669–1691. MR 2679727, DOI 10.3934/dcds.2010.28.1669
Jie Shen, Jie Xu, and Jiang Yang, A new class of efficient and robust energy stable schemes for gradient flows, SIAM Rev. 61 (2019), no. 3, 474–506. MR 3989239, DOI 10.1137/17M1150153
Huailing Song and Chi-Wang Shu, Unconditional energy stability analysis of a second order implicit-explicit local discontinuous Galerkin method for the Cahn-Hilliard equation, J. Sci. Comput. 73 (2017), no. 2-3, 1178–1203. MR 3719623, DOI 10.1007/s10915-017-0497-5
Chuanju Xu and Tao Tang, Stability analysis of large time-stepping methods for epitaxial growth models, SIAM J. Numer. Anal. 44 (2006), no. 4, 1759–1779. MR 2257126, DOI 10.1137/050628143
Jinchao Xu, Yukun Li, Shuonan Wu, and Arthur Bousquet, On the stability and accuracy of partially and fully implicit schemes for phase field modeling, Comput. Methods Appl. Mech. Engrg. 345 (2019), 826–853. MR 3892022, DOI 10.1016/j.cma.2018.09.017
J. Zhu, L.-Q. Chen, J. Shen, and V. Tikare, Coarsening kinetics from a variable-mobility Cahn-Hilliard equation: application of a semi-implicit Fourier spectral method, Phys. Rev. E (3) 60 (1999), 3564–3572.