Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Skip to main content
SpringerLink
Log in

A convex splitting method for the time-dependent Ginzburg-Landau equation

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

Abstract

In this paper, we develop a convex splitting algorithm for the time-dependent Ginzburg-Landau equation, which can preserve both the energy stability and maximum bound principle. The basic idea of the convex splitting method is to decompose the energy functional into the convex part and the concave part. The term corresponding to the convex part of the equation is implicitly treated, and the concave part is explicitly processed. The backward Euler time discretizing method is chosen for the time-dependent Ginzburg-Landau equation. The theoretical analysis proves that the convex splitting method can preserve the maximum bound principle and energy stability. The numerical results show that the numerical algorithm is stable.

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
EUR 32.99 /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 excludes VAT (USA)
Tax calculation will be finalised during checkout.

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

Similar content being viewed by others

Data availability

All data generated or analyzed during this study are included in this published article.

References

  1. Chen, W., Conde, S., Wang, C., Wang, X., Wise, S.M.: A linear energy stable scheme for a thin film model without slope selection. J. Sci. Comput. 52, 546–562 (2012)

    Article  MathSciNet  Google Scholar 

  2. Chen, Z., Dai, S.: Adaptive Galerkin methods with error control for a dynamical Ginzburg-Landau model in superconductivity. SIAM J. Numer. Anal. 38, 1961–1985 (2001)

    Article  MathSciNet  Google Scholar 

  3. Chen, Z., Hoffmann, K.: Numerical studies of a non-stationary Ginzburg-Landau model for superconductivity. Adv. Math. Sci. Appl. 5(2), 363–389 (1995)

    MathSciNet  Google Scholar 

  4. Chen, Z., Hoffmann, K., Liang, J.: On a non-stationary Ginzburg-Landau superconductivity model. Math. Method. Appl. Sci. 16, 855–875 (1993)

    Article  Google Scholar 

  5. Du, Q.: Global existence and uniqueness of solutions of the time-dependent Ginzburg-Landau model for superconductivity. Appl. Anal. 53, 1–17 (1994)

    Article  MathSciNet  Google Scholar 

  6. Du, Q.: Finite element methods for the time-dependent Ginzburg-Landau model of superconductivity. Comput. Math. Appl. 27, 119–133 (1994)

    Article  MathSciNet  Google Scholar 

  7. Eyre, D.: Unconditionally gradient stable time marching the Cahn-Hilliard equation. MRS Online Proceedings Library 529, 39–46 (1998)

    Article  MathSciNet  Google Scholar 

  8. Furihata, D.: A stable and conservative finite difference scheme for the Cahn-Hilliard equation. Numer. Math. 87, 675–699 (2001)

    Article  MathSciNet  Google Scholar 

  9. Ganesh, M., Thompson, T.: A spectrally accurate algorithm and analysis for a Ginzburg-Landau model on superconducting surfaces. Multiscale Model. Sim. 16, 78–105 (2018)

    Article  MathSciNet  Google Scholar 

  10. Gao, H., Ju, L., Xie, W.: A stabilized semi-implicit Euler gauge-invariant method for the time-dependent Ginzburg-Landau equations. J. Sci. Comput. 80, 1083–1115 (2019)

    Article  MathSciNet  Google Scholar 

  11. Gao, H., Li, B., Sun, W.: Optimal error estimates of linearized Crank-Nicolson Galerkin FEMs for the time-dependent Ginzburg-Landau equations in superconductivity. SIAM J. Numer. Anal. 52, 1183–1202 (2014)

    Article  MathSciNet  Google Scholar 

  12. Gao, H., Sun, W.: An efficient fully linearized semi-implicit Galerkin-mixed FEM for the dynamical Ginzburg-Landau equations of superconductivity. J. Comput. Phys. 294, 329–345 (2015)

    Article  MathSciNet  Google Scholar 

  13. Gao, H., Sun, W.: Anew mixed formulation and efficient numerical solution of Ginzburg-Landau equations under the temporal gauge. SIAM J. Sci. Comput. 38, A1339–A1357 (2016)

    Article  Google Scholar 

  14. Gao, H., Sun, W.: Analysis of linearized Galerkin-mixed FEMs for the time-dependent Ginzburg-Landau equations of superconductivity. Adv. Comput. Math. 44, 923–949 (2018)

    Article  MathSciNet  Google Scholar 

  15. Gomez, H., Hughes, T.: Provably unconditionally stable, second-order time-accurate, mixed variational methods for phase-field models. J. Comput. Phys. 230, 5310–5327 (2011)

    Article  MathSciNet  Google Scholar 

  16. Gor’kov, L., Éliashberg, G.: Generalization of the Ginburg-Landau equations for non-stationary problems in the case of alloys with paramagnetic impurities. Soviet Phys. JETP 27, 328–334 (1968)

    Google Scholar 

  17. Guillén-González, F., Tierra, G.: On linear schemes for a Cahn-Hilliard diffuse interface model. J. Comput. Phys. 234, 140–171 (2013)

    Article  MathSciNet  Google Scholar 

  18. Hecht, F.: New development in FreeFem++. J. Numer. Math. 20, 251–65 (2012)

    Article  MathSciNet  Google Scholar 

  19. Heywood, J., Rannacher, R.: Finite-element approximation of the nonstationary Navier-Stokes problem. part IV: error analysis for second-order time discretization. SIAM J. Numer. Anal. 27, 353–384 (1990)

    Article  MathSciNet  Google Scholar 

  20. Li, B., Wang, K., Zhang, Z.: A Hodge decomposition method for dynamic Ginzburg-Landau equations in nonsmooth domains-a second approach. Commun. Comput. Phys. 28, 768–802 (2020)

    Article  MathSciNet  Google Scholar 

  21. Li, B., Zhang, Z.: Mathematical and numerical analysis of the time-dependent Ginzburg-Landau equations in nonconvex polygons based on Hodge decomposition. Math. Comput. 86, 1579–1608 (2017)

    Article  MathSciNet  Google Scholar 

  22. Li, B., Zhang, Z.: A new approach for numerical simulation of the time-dependent Ginzburg-Landau equations. J. Comput. Phys. 303, 238–250 (2015)

    Article  MathSciNet  Google Scholar 

  23. Mu, M., Huang, Y.: An alternating Crank-Nicolson method for decoupling the Ginzburg-Landau equations. SIAM J. Numer. Anal. 35, 1740–1761 (1998)

    Article  MathSciNet  Google Scholar 

  24. Nochetto, R., Pyo, J.: Optimal relaxation parameter for the Uzawa method. Numer. Math. 98, 695–702 (2004)

    Article  MathSciNet  Google Scholar 

  25. Shen, J., Xu, J., Yang, J.: A new class of efficient and robust energy stable schemes for gradient flows. SIAM Rev. 61, 474–506 (2019)

    Article  MathSciNet  Google Scholar 

  26. Shen, J., Zhang, X.: Discrete maximum principle of a high order finite difference scheme for a generalized Allen-Cahn equation. (2021) arXiv:2104.11813

  27. Si, Z.: A generalized scalar auxiliary variable method for the time-dependent Ginzburg-Landau equations. arxiv, (2022) arXiv:2210.08425

  28. Tan, Z., Tang, H.: A general class of linear unconditionally energy stable schemes for the gradient flows. J. Comput. Phys. 464, 111372 (2022)

    Article  MathSciNet  Google Scholar 

  29. Tang, T., Qiao, Z.: Efficient numerical methods for phase-field equations (in Chinese). Sci Sin Math. 50, 1–20 (2020)

    Google Scholar 

  30. Wu, C., Sun, W.: Analysis of Galerkin FEMs for mixed formulation of time-dependent Ginzburg-Landau equations under temporal gauge. SIAM J. Numer. Anal. 56, 1291–1312 (2018)

    Article  MathSciNet  Google Scholar 

  31. Zhang, Y., Shen, J.: A generalized SAV approach with relaxation for dissipative systems. J. Comput. Phys. 464, 111311 (2022)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors are very much indebted to the referees for their constructive suggestions and insightful comments, which greatly improved the original manuscript of this paper.

Funding

This work is supported by the National Natural Science Foundation of China (No. 11971152 & 12126318).

Author information

Authors and Affiliations

  1. School of Mathematics and Information Science, Henan Polytechnic University, 454003, Jiaozuo, People’s Republic of China

    Yunxia Wang & Zhiyong Si

Authors
  1. Yunxia Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhiyong Si
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

YW: writing—original draft preparation, analysis.

ZS: methodology, software, reviewing.

Corresponding author

Correspondence to Zhiyong Si.

Ethics declarations

Ethical approval

Not applicable

Conflict of interest

The authors declare no competing interests.

Additional information

Publisher's Note

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

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.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, Y., Si, Z. A convex splitting method for the time-dependent Ginzburg-Landau equation. Numer Algor 96, 999–1017 (2024). https://doi.org/10.1007/s11075-023-01672-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-023-01672-0

Keywords

Search

Navigation