Skip to main content
SpringerLink
Log in

A new linearized maximum principle preserving and energy stability scheme for the space fractional Allen-Cahn equation

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

Abstract

In this paper, a numerical method is proposed to solve the space fractional Allen-Cahn equation. Based on Crank-Nicolson method for time discretization and second-order weighted and shifted Grünwald difference formula for spatial discretization, we present a new linearized two-level scheme, where the nonlinear term is handled by Newton linearized technology. And we only need to solve a linear system at each time level. Then, the unique solvability of the numerical scheme is given. Under the appropriate assumptions of time step, the discrete maximum principle and energy stability of the numerical scheme are proved. Furthermore, we give a detailed error analysis, which reflects that the temporal and spatial convergence orders are both second order. At last, some numerical experiments show that the proposed method is reasonable and effective.

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

Access this article

Log in via an institution

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

Similar content being viewed by others

Data availability

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

References

  1. Allen, S.M., Cahn, J.W.: A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall. 27, 1085–1095 (1979)

    Article  Google Scholar 

  2. Du, Q., Liu, C., Wang, X.: A phase field approach in the numerical study of the elastic bending energy for vesicle membranes. J. Comput. Phys. 198, 450–468 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  3. Evans, L.C., Soner, H.M., Souganidis, P.E.: Phase transitions and generalized motion by mean curvature. Commun. Pure Appl. Math. 45, 1097–1123 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  4. Evans, L.C., Spruck, J.: Motion of level sets by mean curvature. I. J. Differ. Geom. 33, 635–681 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  5. Tang, T., Yang, J.: Implicit-explicit scheme for the Allen-Cahn equation preserves the maximum principle. J. Comput. Math. 34, 471–781 (2016)

    MathSciNet  MATH  Google Scholar 

  6. Hou, T., Xiu, D., Jiang, W.: A new second-order maximum-principle preserving finite difference scheme for Allen-Cahn equations with periodic boundary conditions. Appl. Math. Lett. 104, 106265 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  7. Feng, X., Prohl, A.: Numerical analysis of the Allen-Cahn equation and approximation for mean curvature flows. Numer. Math. 94, 33–65 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  8. Feng, X., Song, H., Tang, T., Yang, J.: Nonlinear stability of the implicit-explicit methods for the Allen-Cahn equation. Inverse Probl. Imaging. 7, 679–695 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  9. Feng, X., Tang, T., Yang, J.: Stabilized Crank-Nicolson/Adams-Bashforth schemes for phase field models. East Asian J. Appl. Math. 3, 59–80 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hou, T., Leng, H.: Numerical analysis of a stabilized Crank-Nicolson/Adams-Bashforth finite difference scheme for Allen-Cahn equations. Appl. Math. Lett. 102, 106150 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  11. Shen, J., Tang, T., Yang, J.: On the maximum principle preserving schemes for the generalized Allen-Cahn equation. Commun. Math. Sci. 14, 1517–1534 (2016)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  13. Tan, Z., Zhang, C.: The discrete maximum principle and energy stability of a new second-order difference scheme for Allen-Cahn equations. Appl. Numer. Math. 166, 227–237 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  14. Yang, X.: Error analysis of stabilized semi-implicit method of Allen-Cahn equation. Discrete Contin. Dyn. Syst., Ser. B 11, 1057–1070 (2009)

    MathSciNet  MATH  Google Scholar 

  15. Liao, H., Tang, T., Zhou, T.: On energy stable, maximum-principle preserving, second-order BDF scheme with variable steps for the Allen-Cahn equation. SIAM J. Numer. Anal. 58, 2294–2314 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  16. Yang, Y., Chen, Y., Huang, Y., et al.: Spectral collocation method for the time-fractional diffusion-wave equation and convergence analysis. Comput. Math. Appl. 73(6), 1218–1232 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  17. Yang, Y., Huang, Y., Zhou, Y.: Numerical solutions for solving time fractional Fokker-Planck equations based on spectral collocation methods. J. Comput. Appl. Math. 339, 389–404 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  18. Yang, Y., Huang, Y., Zhou, Y.: Numerical simulation of time fractional Cable equations and convergence analysis. Numer. Meth. Part. D. E. 34(5), 1556–1576 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  19. Yang, Y., Wang, J., Zhang, S., et al.: Convergence analysis of space-time Jacobi spectral collocation method for solving time-fractional schrödinger equations. Appl. Math. Comput. 387, 124489 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  20. Yang, Y., Chen, Y., Huang, Y.: Convergence analysis of the Jacobi spectral-collocation method for fractional integro-differential equations. Acta. Math. Sci. 34(3), 673–690 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  21. Nec, Y., Nepomnyashchy, A., Golovin, A.: Front-type solutions of fractional Allen-Cahn equation. Physica D. 237, 3237–3251 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  22. Hou, T., Tang, T., Yang, J.: Numerical analysis of fully discretized Crank-Nicolson scheme for fractional-in-space Allen-Cahn equations. J. Sci. Comput. 72, 1214–1231 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  23. He, D., Pan, K., Hu, H.: A spatial fourth-order maximum principle preserving operator splitting scheme for the multi-dimensional fractional Allen-Cahn equation. Appl. Numer. Math. 151, 44–63 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  24. Zhang, H., Yan, J., Qian, X., et al.: On the preserving of the maximum principle and energy stability of high-order implicit-explicit Runge-Kutta schemes for the space-fractional Allen-Cahn equation. Numer. Algorithms. 88(3), 1309–1336 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  25. Chen, H., Sun, H.: A dimensional splitting exponential time differencing scheme for multidimensional fractional Allen-Cahn equations. J. Sci. Comput. 87, 1–25 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  26. Chen, H., Sun, H.: Second-order maximum principle preserving Strang’s splitting schemes for anisotropic fractional Allen-Cahn equations. Numer. Algorithms. 1–23 (2021)

  27. Bu, L., Mei, L., Hou, Y.: Stable second-order schemes for the space-fractional Cahn-Hilliard and Allen-Cahn equations. Comput. Math. Appl. 78(11), 3485–3500 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  28. Tian, W., Zhou, H., Deng, W.: A class of second order difference approximations for solving space fractional diffusion equations. Math. Comput. 84, 1703–1727 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  29. Wu, F., Cheng, X., Li, D., et al.: A two-level linearized compact ADI scheme for two-dimensional nonlinear reaction-diffusion equations. Comput. Math. Appl. 75, 2835–2850 (2018)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Funding

Yin Yang is supported by the National Natural Science Foundation of China Project (12071402), the National Key Research and Development Program of China (2020YFA0713503), the Project of Scientific Research Fund of the Hunan Provincial Science and Technology Department (2020JJ2027). Biao Zhang is supported by the Postgraduate Scientific Research Innovation Project of Hunan Province (CX20210594).

Author information

Authors and Affiliations

  1. School of Mathematics and Computational Science, Xiangtan University, Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education, Xiangtan, 411105, Hunan, China

    Biao Zhang

  2. School of Mathematics and Computational Science, Xiangtan University, National Center for Applied Mathematics in Hunan, Hunan International Scientific and Technological Innovation Cooperation Base of Computational Science, Xiangtan, 411105, Hunan, China

    Yin Yang

Authors
  1. Biao Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yin Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yin Yang.

Ethics declarations

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 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

Zhang, B., Yang, Y. A new linearized maximum principle preserving and energy stability scheme for the space fractional Allen-Cahn equation. Numer Algor 93, 179–202 (2023). https://doi.org/10.1007/s11075-022-01411-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-022-01411-x

Keywords

Search

Navigation