[go: up one dir, main page]
More Web Proxy on the site http://driver.im/ Skip to main content
Springer Nature Link
Log in

Energy dissipation–preserving time-dependent auxiliary variable method for the phase-field crystal and the Swift–Hohenberg models

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

In this study, we develop first- and second-order time-accurate energy stable methods for the phase-field crystal equation and the Swift–Hohenberg equation with quadratic-cubic non-linearity. Based on a new Lagrange multiplier approach, the first-order time-accurate schemes dissipate the original energy in a time-discretized version, which are different from the modified energy laws obtained by the invariant energy quadratization (IEQ) and the scalar auxiliary variable (SAV) methods. Moreover, the proposed schemes do not require the bounded-from-below restriction which is necessary in the IEQ or SAV approach. We rigorously prove the energy dissipations of first- and second-order accurate methods with respect to the original energy and pseudo-energy in the time-discretized versions, respectively. An efficient algorithm is used to decouple the resulting weakly coupled systems. In one time iteration, only two linear systems with constant coefficients and one non-linear algebraic equation need to be solved. Finally, the accuracy, stability and practicability of the proposed methods are validated by intensive numerical tests.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
£29.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17

Similar content being viewed by others

References

  1. Kim, J.: Phase-field models for multi-component fluid flows. Commun. Comput. Phys. 12(3), 613–661 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  2. Mukherjee, D., Larsson, H., Odqvist, J.: Phase field modelling of diffusion induced grain boundary migration in binary alloys. Comput. Mater. Sci. 109914, 184 (2020)

    Google Scholar 

  3. Kubendran Amos, P.G., Schoof, E., Santoki, J., Schneider, D., Nestler, B.: Limitations of preserving volume in Allen–Cahn framework for microstructural analysis. Comput. Mater. Sci. 109388, 173 (2020)

    Google Scholar 

  4. Lee, S.: Mathematical model of contractile ring-driven cytokinesis in a three-dimensional domain. Bull. Math. Biol. 80(3), 583–597 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chen, Y., Wise, S.M., Shenoy, V.B., Lowengrub, J.S.: A stable scheme for a nonlinear, multiphase tumor growth model with an elastic membrane. Int. J. Numer. Meth. Biomed. Engng. 30, 726–754 (2014)

    Article  MathSciNet  Google Scholar 

  6. Eyre, D.J.: Unconditionally gradient stable time marching the Cahn–Hilliard equation. In: Computational and Mathematical Models of Microstructural Evolution (San Francisco, CA, 1998), Mater. Res. Soc. Sympos. Proc. Warrendale, PA, 529, 39–46 (1998)

  7. Hu, Z., Wise, S.M., Wang, C., Lowengrub, J.S.: Stable and efficient finite-difference nonlinear-multigrid schemes for the phase field crystal equation. J. Comput. Phys. 228, 5323–5339 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. Wise, S.M., Wang, C., Lowengrub, J.S.: An energy-stable and convergent finite-difference scheme for the phase field crystal equation. SIAM. J. Numer. Anal. 44, 2269–2288 (2009)

    Article  MATH  Google Scholar 

  9. Dong, L., Feng, W., Wang, C., Wise, S.M., Zhang, Z.: Convergence analysis and numerical implementation of a second order numerical scheme for the three-dimensional phase field crystal equation. Comput. Math. Appl. 75(6), 1912–1928 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  10. Yan, Y., Chen, W., Wang, C., Wise, S.M.: A second-order energy stable BDF numerical scheme for the Cahn–Hilliard equation. Commun. Comput. Phys. 23(2), 572–602 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  11. Shen, J., Yang, X.: Numerical approximations of Allen–Cahn and Cahn–Hilliard equations. Discrete Contin. Dyn. Syst. 28(4), 1669–1691 (2010)

    MathSciNet  MATH  Google Scholar 

  12. Pei, S., Hou, Y., You, B.: A linearly second-order energy stable scheme for the phase field crystal model. Appl. Numer. Math. 140, 134–164 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  13. Zhao, S., Xiao, X., Feng, X.: A efficient time adaptivity based on chemical potential for surface Cahn–Hilliard equation using finite element approximation. Appl. Math. Comput. 124901, 369 (2020)

    MathSciNet  MATH  Google Scholar 

  14. Liu, Z., Li, X.: Efficient modified techniques of invariant energy quadratization approach for gradient flows. Appl. Math. Lett. 98, 206–214 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  15. Yang, J., Kim, J.: A variant of stabilized-scalar auxiliary variable (s-SAV) approach for a modified phase-field surfactant model. Comput. Phys. Commun. 107825, 261 (2021)

    MathSciNet  Google Scholar 

  16. Li, Q., Mei, L.: Efficient, decoupled, and second-order unconditionally energy stable numerical schemes for the coupled Cahn–Hilliard system in copolymer/homopolymer mixtures. Comput. Phys. Commun. 107290, 260 (2021)

    MathSciNet  Google Scholar 

  17. Xu, C., Chen, C., Yang, X.: Efficient, non-iterative, and decoupled numerical scheme for a new modified binary phase-field surfactant system. Numer. Algor. 86, 863–885 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  18. Elder, K.R., Katakowski, M., Haataja, M., Grant, M.: Modeling elasticity in crystal growth. Phys. Rev. Lett. 88(24), 245701 (2002)

    Article  Google Scholar 

  19. Wang, C., Wise, S.M., Lowengrub, J.S.: An energy-stable and convergent finite-difference scheme for the phase field crystal equation. SIAM. J. Numer. Anal. 47(3), 2269–2288 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. Lee, H.G., Shin, J., Lee, J.-Y.: First and second order operator splitting methods for the phase field crystal equation. J. Comput. Phys. 299, 82–91 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  21. Shin, J., Lee, H.G., Lee, J.-Y.: First and second order numerical methods based on a new convex splitting for phase-field crystal equation. J. Comput. Phys. 327, 519–542 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  22. Yang, X., Han, D.: Linearly first- and second-order, unconditionally energy stable schemes for the phase field crystal model. J. Comput. Phys. 330, 1116–1134 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  23. Liu, Z., Li, X.: Efficient modified stabilized invariant energy quadratization approaches for phase-field crystal equation. Numer. Algor. 85, 107–132 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  24. Wang, L., Huang, Y., Jiang, K.: Error analysis of SAV finite element method to phase field crystal model. Numer. Math. Theor. Meth. Appl. 13, 372–399 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  25. Li, X., Shen, J.: Stability and error estimates of the SAV Fourier-spectral method for the phase-field crystal equation. Adv. Comput. Math. 46, 1–20 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  26. Swift, J., Hohenberg, P.C.: Hydrodynamics fluctuations at the convective instability. Phys. Rev. A 15, 319–328 (1977)

    Article  Google Scholar 

  27. Cross, M.C., Hohenberg, P.C.: Pattern formation outside of equilibrium. Rev. Modern Phys. 65, 851–1112 (1993)

    Article  MATH  Google Scholar 

  28. Hutt, A., Longtin, A., Schimansky-Geier, L.: Additive noise-induced Turing transition in spatial systems with application to neural fields and the Swift–Hohenberg equation. Physica D 237, 755–773 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  29. Lee, H.G.: Numerical simulation of pattern formation on surfaces using an efficient linear second-order method. Symmetry 11(8), 1010 (2019)

    Article  Google Scholar 

  30. Gomez, H., Nogueira, X.: A new space-time discretization for the Swift–Hohenberg equation that strictly respects the Lyapunov functional. Commun. Nonlinear Sci. Numer. Simul. 17, 4930–4946 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  31. Su, J., Fang, W., Yu, Q., Li, Y.: Numerical simulation of Swift–Hohenberg equation by the fourth-order compact scheme. Comput. Appl. Math. 38, 54 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  32. Lee, H.G.: An energy stable method for the Swift–Hohenberg equation with quadratic-cubic nonlinearity. Comput. Methods Appl. Mech. Engrg. 343, 40–51 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  33. Wang, J., Zhai, S.: A fast and efficient numerical algorithm for the nonlocal conservative Swift–Hohenberg equation. Math. Probl. Eng. 2020, 7012483 (2020)

    MathSciNet  Google Scholar 

  34. Liu, H., Yin, P.: Unconditionally energy stable DG schemes for the Swift–Hohenberg equation. J. Sci. Comput. 81, 789–819 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  35. Liu, H., Yin, P.: Energy stable Runge–Kutta discontinuous Galerkin schemes for fourth order gradient flows. arXiv preprint arXiv:2101.00152(2021)

  36. Liu, H., Yin, P.: On the SAV-DG method for a class of fourth order gradient flows. arXiv preprint arXiv:2008.11877 (2020)

  37. Cheng, Q., Liu, C., Shen, J.: A new Lagrange multiplier approach for gradient flows. Comput. Methods Appl. Mech. Engrg. 113070, 367 (2020)

    MathSciNet  MATH  Google Scholar 

  38. Yoon, S., Jeong, D., Lee, C., Kim, H., Lee, H.G., Kim, J.: Fourier-spectral method for the phase-field equations. Mathematics 8 (8), 1385 (2020)

    Article  Google Scholar 

  39. Li, X., Rui, H., Liu, Z.: Two alternating direction implicit spectral methods for two-dimensional distributed-order differential equation. Numer. Algor. 82(1), 321–347 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  40. Lee, H.G., Kim, J.: A simple and efficient finite difference method for the phase-field crystal equation on curved surface. Comput. Methods Appl. Mech. Engrg. 307, 32–43 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  41. Zhang, J., Yang, X.: Efficient second order unconditionally stable time marching numerical scheme for a modified phase-field crystal model with a strong nonlinear vacancy potential. Comput. Phys. Commun. 106860, 245 (2019)

    MathSciNet  Google Scholar 

  42. Liu, Z.: Novel energy stable schemes for Swift–Hohenberg model with quadratic-cubic nonlinearity based on the H− 1-gradient flow approach. Numer. Algor. https://doi.org/10.1007/s11075-020-00981-y (2020)

  43. Xia, B., Mei, C., Yu, Q., Li, Y.: A second order unconditionally stable scheme for the modified phase field crystal model with elastic interaction and stochastic noise effect. Comput. Methods Appl. Mech. Engrg. 363, 112795 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  44. Liu, Z., Li, X.: Two fast and efficient linear semi-implicit approaches with unconditional energy stability for nonlocal phase field crystal equation. Appl. Numer. Math. 150, 491–506 (2020)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Funding

J. Yang is supported by the China Scholarship Council (201908260060). The corresponding author (J.S. Kim) was supported by the Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education(NRF-2019R1A2C1003053).

Author information

Authors and Affiliations

  1. Department of Mathematics, Korea University, Seoul, 02841, Republic of Korea

    Junxiang Yang & Junseok Kim

Authors
  1. Junxiang Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Junseok Kim
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Junseok Kim.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yang, J., Kim, J. Energy dissipation–preserving time-dependent auxiliary variable method for the phase-field crystal and the Swift–Hohenberg models. Numer Algor 89, 1865–1894 (2022). https://doi.org/10.1007/s11075-021-01176-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-021-01176-9

Keywords

Search

Navigation