Abstract
In this paper, we design and analyze an unconditionally energy-stable, second-order-in-time, finite element scheme for the Swift–Hohenberg equation. We prove rigorously that our scheme is unconditionally uniquely solvable and unconditionally energy stable. We also give the boundedness of discrete phase variable for any time and space mesh sizes. Numerical tests are presented to validate the accuracy and energy stability of our scheme.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data Availability
No data was used for the research described in the article.
References
Abbaszadeh M, Khodadadian A, Parvizi M, Dehghan M, Heitzinger C (2019) A direct meshless local collocation method for solving stochastic Cahn–Hilliard–Cook and stochastic Swift–Hohenberg equations. Eng Anal Boundary Elem 98:253–264. https://doi.org/10.1016/j.enganabound.2018.10.021
Baskaran A, Hu Z, Lowengrub JS, Wang C, Wise SM, Zhou P (2013) Energy stable and efficient finite-difference nonlinear multigrid schemes for the modified phase field crystal equation. J Comput Phys 250:270–292
Baskaran A, Lowengrub JS, Wang C, Wise SM (2013) Convergence analysis of a second order convex splitting scheme for the modified phase field crystal equation. SIAM J Numer Anal 51(5):2851–2873. https://doi.org/10.1137/120880677
Cheng K, Wang C, Wise SM (2019) An energy stable BDF2 Fourier pseudo-spectral numerical scheme for the square phase field crystal equation. Commun Comput Phys 26(5):1335–1364
Cheng M, James AW (2008) An efficient algorithm for solving the phase field crystal model. J Comput Phys 227(12):6241–6248. https://doi.org/10.1016/j.jcp.2008.03.012
Dehghan M, Abbaszadeh M (2017) The meshless local collocation method for solving multi-dimensional Cahn–Hilliard, Swift–Hohenberg and phase field crystal equations. Eng Anal Boundary Elem 78:49–64. https://doi.org/10.1016/j.enganabound.2017.02.005
Dehghan M, Abbaszadeh M, Khodadadian A, Heitzinger C (2019) Galerkin proper orthogonal decomposition-reduced order method (POD-ROM) for solving generalized Swift–Hohenberg equation. Int J Numer Meth Heat Fluid Flow 29(8):2642–2665. https://doi.org/10.1108/HFF-11-2018-0647
Dehghan M, Gharibi Z, Eslahchi MR (2022) Unconditionally energy stable C0-virtual element scheme for solving generalized Swift–Hohenberg equation. Appl Numer Math 178:304–328. https://doi.org/10.1016/j.apnum.2022.03.013
Diegel AE (2015) Numerical analysis of convex splitting schemes for Cahn-Hilliard and coupled Cahn-Hilliard-fluid-flow equations. Ph.D. Thesis, University of Tennessee
Diegel AE, Feng XH, Wise SM (2015) Analysis of a mixed finite element method for a Cahn–Hilliard–Darcy–Stokes system. SIAM J Numer Anal 53(1):127–152
Elsey M, Wirth B (2013) A simple and efficient scheme for phase field crystal simulation. ESAIM Math Model Numer Anal 47(5):1413–1432
Gomez H, Nogueira X (2012) A new space-time discretization for the Swift–Hohenberg equation that strictly respects the Lyapunov functional. Commun Nonlinear Sci Numer Simul 17(12):4930–4946
Hu Z, Wise SM, Wang C, Lowengrub JS (2009) Stable and efficient finite-difference nonlinear-multigrid schemes for the phase field crystal equation. J Comput Phys 228(15):5323–5339
Lee HG (2017) A semi-analytical Fourier spectral method for the Swift–Hohenberg equation. Comput Math Appl 74(8):1885–1896
Lee HG (2019) An energy stable method for the Swift-Hohenberg equation with quadratic-cubic nonlinearity. Comput Methods Appl Mech Eng 343:40–51
Lee HG (2020) A new conservative Swift–Hohenberg equation and its mass conservative method. J Comput Appl Math 375:112815. https://doi.org/10.1016/j.cam.2020.112815
Li Y, Kim J (2017) An efficient and stable compact fourth-order finite difference scheme for the phase field crystal equation. Comput Methods Appl Mech Eng 319:194–216. https://doi.org/10.1016/j.cma.2017.02.022
Qi L, Hou Y (2021) A second order energy stable BDF numerical scheme for the Swift–Hohenberg equation. J Sci Comput 88(3):1–25
Qi L, Hou Y (2022) Error analysis of first- and second-order linear, unconditionally energy-stable schemes for the Swift–Hohenberg equation. Comput Math Appl 127:192–212. https://doi.org/10.1016/j.camwa.2022.10.007
Qi L, Hou Y (2022) An unconditionally energy-stable linear Crank–Nicolson scheme for the Swift–Hohenberg equation. Appl Numer Math 181:46–58. https://doi.org/10.1016/j.apnum.2022.05.018
Qi L, Hou Y (2022c) Numerical analysis of a second-order mixed finite element method for the Swift-Hohenberg equation. preprint
Qi L, Hou Y (2022) Error estimate of a stabilized second-order linear predictor-corrector scheme for the Swift-Hohenberg equation. Appl Math Lett 127:107836
Qi L, Hou Y (2023) Error estimates for the Scalar Auxiliary Variable (SAV) schemes to the modified phase field crystal equation. J Comput Appl Math 417:114579. https://doi.org/10.1016/j.cam.2022.114579
Swift JB, Hohenberg PC (1977) Hydrodynamic fluctuations at the convective instability. Phys Rev A 15(1):319–328
Wang C, Wise SM (2011) An energy stable and convergent finite-difference scheme for the modified phase field crystal equation. SIAM J Numer Anal 49(3):945–969. https://doi.org/10.1137/090752675
Wise SM, Wang C, Lowengrub JS (2009) An energy-stable and convergent finite-difference scheme for the phase field crystal equation. SIAM J Numer Anal 47(3):2269–2288. https://doi.org/10.1137/080738143
Yang X, Han D (2017) Linearly first- and second-order, unconditionally energy stable schemes for the phase field crystal model. J Comput Phys 330:1116–1134
Zhai S, Weng Z, Feng X, He Y (2021) Stability and error estimate of the operator splitting method for the phase field crystal equation. J Sci Comput 86:1
Zhang Z, Ma Y (2016) On a large time-stepping method for the Swift-Hohenberg equation. Adv Appl Math Mech 8(6):992–1003
Acknowledgements
This work was supported by the National Natural Science Foundation of China (NSFC) (No. 11971378).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflicts of Interests
The authors declared that they have no conflicts of interest to this work.
Additional information
Communicated by Frederic Valentin.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Subsidized by National Natural Science Foundation of China (NSFC) (Grant No.11971378).
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Qi, L., Hou, Y. An energy-stable second-order finite element method for the Swift–Hohenberg equation. Comp. Appl. Math. 42, 5 (2023). https://doi.org/10.1007/s40314-022-02144-2
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40314-022-02144-2