Skip to main content
SpringerLink
Log in

Highly efficient schemes for time-fractional Allen-Cahn equation using extended SAV approach

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

Abstract

In this paper, we propose and analyze high-order efficient schemes for the time-fractional Allen-Cahn equation. The proposed schemes are based on the L1 discretization for the time-fractional derivative and the extended scalar auxiliary variable (SAV) approach developed very recently to deal with the nonlinear terms in the equation. The main contributions of the paper consist of (1) constructing first- and higher order unconditionally stable schemes for different mesh types, and proving the unconditional stability of the constructed schemes for the uniform mesh; (2) carrying out numerical experiments to verify the efficiency of the schemes and to investigate the coarsening dynamics governed by the time-fractional Allen-Cahn equation. In particular, the influence of the fractional order on the coarsening behavior is carefully examined. Our numerical evidence shows that the proposed schemes are more robust than the existing methods, and their efficiency is less restricted to particular forms of the nonlinear potentials.

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

Similar content being viewed by others

References

  1. Acosta, G., Bersetche, F.: Numerical approximations for a fully fractional Allen-Cahn equation, pp. 1–32 arXiv:http://arxiv.org/abs/1903.08964 (2019)

  2. Ainsworth, M., Mao, Z.: Analysis and approximation of a fractional Cahn-Hilliard equation. SIAM J. Numer. Anal. 55(4), 1689–1718 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ainsworth, M., Mao, Z.: Well-posedness of the Cahn-Hilliard equation with fractional free energy and its Fourier Galerkin approximation. Chaos Soliton Fract. 102, 264–273 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  4. Akagi, G., Schimperna, G., Segatti, A.: Fractional Cahn-Hilliard, Allen-Cahn and porous medium equations. J. Differ. Equ. 261(6), 2935–2985 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  5. Alikhanov, A.A.: A priori estimates for solutions of boundary value problems for fractional-order equations. Differ. Equ. 46(5), 660–666 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Alikhanov, A.A.: A new difference scheme for the time fractional diffusion equation. J. Comput. Phys. 280(C), 424–438 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  7. Allen, S.M., Cahn, J.W.: A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall. Mater. 27(6), 1085–1095 (1979)

    Article  Google Scholar 

  8. Anderson, D.M., Mcfadden, G.B., Wheeler, A.A.: Diffuse-interface methods in fluid mechanics. Annu. Rev. Fluid Mech. 30, 139–165 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  9. Baffet, D., Hesthaven, J.S.: High-order accurate adaptive kernel compression time-stepping schemes for fractional differential equations. J. Sci. Comput. 72(3), 1169–1195 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  10. Baffet, D., Hesthaven, J.S.: A kernel compression scheme for fractional differential equations. SIAM J. Numer. Anal. 55(2), 496–520 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  11. Cahn, J.W., Hilliard, JE: Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phys. 28, 258–267 (1958)

    Article  MATH  Google Scholar 

  12. Chen, L., Shen, J.: Applications of semi-implicit Fourier-spectral method to phase field equations. Comput. Phys. Commun. 108(2–3), 147–158 (1998)

    Article  MATH  Google Scholar 

  13. Chen, L., Zhao, J., Cao, W., Wang, H, Zhang, J.: An accurate and efficient algorithm for the time-fractional molecular beam epitaxy model with slope selection. Comput. Phys. Commun. 245, 106842 (2019)

    Article  MathSciNet  Google Scholar 

  14. Doi, M., Edwards, S.F.: The Theory of Polymer Dynamics, vol. 73. Oxford University Press, Oxford (1988)

    Google Scholar 

  15. Du, Q., Ju, L., Li, X., Qiao, Z.: Stabilized linear semi-implicit schemes for the nonlocal Cahn-Hilliard equation. J. Comput. Phys. 363, 39–54 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  16. Du, Q., Yang, J., Zhou, Z.: Time-fractional Allen-Cahn equations: analysis and numerical methods, pp. 1–24. arXiv:http://arxiv.org/abs/1906.06584v1 (2019)

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

    Article  Google Scholar 

  18. Gao, G., Sun, Z., Zhang, Y.: A finite difference scheme for fractional sub-diffusion equations on an unbounded domain using artificial boundary conditions. J. Comput. Phys. 231(7), 2865–2879 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  19. Gao, G., Sun, Z., Zhang, H.: A new fractional numerical differentiation formula to approximate the caputo fractional derivative and its applications. J. Comput. Phys. 259(2), 33–50 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  20. Gurtin, M.E., Polignone, D.: Two-phase binary fluids and immiscible fluids described by an order parameter. Math. Models Methods Appl. Sci. 6, 815–831 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  21. Hou, D., Azaiez, M., Xu, C.: A variant of scalar auxiliary variable approaches for gradient flows. J. Comput. Phys. 395, 307–332 (2019). https://doi.org/10.1016/j.jcp.2019.05.037

    Article  MathSciNet  MATH  Google Scholar 

  22. Ji, B., Liao, H., Gong, Y., Zhang, L.: Adaptive linear second-order energy stable schemes for time-fractional Allen-Cahn equation with volume constraint, pp. 1–21. arXiv:http://arxiv.org/abs/1909.10936 (2019)

  23. Jiang, S., Zhang, J., Zhang, Q., Zhang, Z: Fast evaluation of the Caputo fractional derivative and its applications to fractional diffusion equations. Commun. Comput. Phys. 21(3), 650–678 (2017)

    Article  MathSciNet  Google Scholar 

  24. Khalid, D.N., Abbas, M., Iqbal, M.K., Baleanu, D.: A numerical investigation of Caputo time fractional Allen-Cahn equation using redefined cubic B-spline functions. Adv. Differ. Equ. 2020(1) (2020)

  25. Leslie, F.M.: Theory of flow phenomena in liquid crystals. Adv. Liq. Cryst. 4, 1–81 (1979)

    Article  Google Scholar 

  26. Li, Z., Wang, H., Yang, D.: A space-time fractional phase-field model with tunable sharpness and decay behavior and its efficient numerical simulation. J Comput. Phys. 347, 20–38 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  27. Liao, H., Tang, T., Zhou, T.: A second-order and nonuniform time-stepping maximum-principle preserving scheme for time-fractional Allen-Cahn equations, pp. 1–22 arXiv:http://arxiv.org/abs/1909.10216 (2019)

  28. Lin, Y., Xu, C.: Finite difference/spectral approximations for the time-fractional diffusion equation. J. Comput. Phys. 225(2), 1533–1552 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  29. Liu, H., Cheng, A., Wang, H., Zhao, J.: Time-fractional Allen-Cahn and Cahn-Hilliard phase-field models and their numerical investigation. Comput. Math. Appl. 76, 1876–1896 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  30. Lv, C., Xu, C.: Improved error estimates of a finite difference/spectral method for time-fractional diffusion equations. Int. J. Numer. Anal. Mod. 12(2), 384–400 (2015)

    MathSciNet  MATH  Google Scholar 

  31. Lv, C., Xu, C.: Error analysis of a high order method for time-fractional diffusion equation. SIAM J. Sci. Comput. 38(5), A2699–A2724 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  32. McCartin, B.J.: On polygonal domains with trigonometric eigenfunctions of the Laplacian under Dirichlet or Neumann boundary conditions. Appl. Math. Sci. 2, 2891–2901 (2008)

    MathSciNet  MATH  Google Scholar 

  33. Shen, J., Xu, J.: Convergence and error analysis for the scalar auxiliary variable (SAV) schemes to gradient flows. SIAM J. Numer Anal. 56(5), 2895–2912 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  34. Shen, J., Xu, J., Yang, J.: The scalar auxiliary variable (SAV) approach for gradient flows. J. Comput. Phys. 353, 407–416 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  35. Shen, J., Xu, J., Yang, J.: A new class of efficient and robust energy stable schemes for gradient flows. SIAM Rev. 61(3), 474–506 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  36. Song, F., Xu, C., Em Karniadakis, G.: A fractional phase-field model for two-phase flows with tunable sharpness: algorithms and simulations. Comput. Methods Appl. Mech. Eng. 305, 376–404 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  37. Sun, Z., Wu, X.: A fully discrete difference scheme for a diffusion-wave system. Appl. Numer. Math. 56(2), 193–209 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  38. Tang, T., Yu, H., Zhou, T.: On energy dissipation theory and numerical stability for time-fractional phase field equations. SIAM J. Sci. Comput. 41(6), A3757–A3778 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  39. Yan, Y., Sun, Z., Zhang, J.: Fast evaluation of the caputo fractional derivative and its applications to fractional diffusion equations: a second-order scheme. Commun. Comput. Phys. 22(4), 1028–1048 (2017)

    Article  MathSciNet  Google Scholar 

  40. Yan, Y., Khan, M., Ford, N.J.: An analysis of the modified L1 scheme for time-fractional partial differential equations with nonsmooth data. SIAM J. Numer. Anal. 56(1), 210–227 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  41. Yang, X.: Linear, first and second-order, unconditionally energy stable numerical schemes for the phase field model of homopolymer blends. J. Comput. Phys. 327, 294–316 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  42. Yue, P., Feng, J., Liu, C., Shen, J.: A diffuse-interface method for simulating two-phase flows of complex fluids. J. Fluid Mech. 515, 293–317 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  43. Zeng, F., Turner, I., Burrage, K.: A stable fast time-stepping method for fractional integral and derivative operators. J. Sci. Comput. 77(1), 283–307 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  44. Zhang, Q., Zhang, J., Jiang, S., Zhang, Z.: Numerical solution to a linearized time fractional KdV equation on unbounded domains. Math. Comput. 87(310) (2018)

  45. Zhao, J., Wang, Q., Yang, X.: Numerical approximations for a phase field dendritic crystal growth model based on the invariant energy quadratization approach. Int. J. Numer. Meth. Eng. 110(3), 279–300 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  46. Zhao, J., Chen, L., Wang, H.: On power law scaling dynamics for time-fractional phase field models during coarsening. Commun. Nonlinear Sci. Numer. Simul. 70, 257–270 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  47. Zhen, G., Lowengrub, J., Wang, C., Wise, S.: Second order convex splitting schemes for periodic nonlocal Cahn-Hilliard and Allen-Cahn equations. J. Comput. Phys. 277, 48–71 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  48. Zhou, X.L., Azaiez, M., Xu, C.J.: Reduced-order modelling for the Allen-Cahn equation based on scalar auxiliary variable approaches. J. Math. Study 52(3), 258–276 (2019)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Funding

The work of D. Hou is supported by NSFC grant 12001248, the NSF of the Jiangsu Higher Education Institutions of China grant BK20201020, the NSF of Universities in Jiangsu Province of China grant 20KJB110013, and the Foundation of Jiangsu Normal University grant 19XSRX022. The second author is supported by NSFC grant 12001238. The third author has received support from NSFC grant 11971408, NNW2018-ZT4A06 project, and NSFC/ANR joint program 51661135011/ANR-16-CE40-0026-01.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Jiangsu Normal University, 221116, Xuzhou, China

    Dianming Hou

  2. School of Intelligent Systems Science and Engineering, Jinan University (Zhuhai Campus), 519070, Zhuhai, China

    Hongyi Zhu

  3. School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Modeling and High Performance Scientific Computing, Xiamen University, 361005, Xiamen, China

    Chuanju Xu

  4. Bordeaux INP, Laboratoire I2M UMR 5295, 33607, Pessac, France

    Chuanju Xu

Authors
  1. Dianming Hou
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hongyi Zhu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Chuanju Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Chuanju Xu.

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

Hou, D., Zhu, H. & Xu, C. Highly efficient schemes for time-fractional Allen-Cahn equation using extended SAV approach. Numer Algor 88, 1077–1108 (2021). https://doi.org/10.1007/s11075-021-01068-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-021-01068-y

Keywords

Mathematics Subject Classification (2010)

Search

Navigation