Stability and error estimates of the SAV Fourier-spectral method for the phase field crystal equation

Abstract

We consider fully discrete schemes based on the scalar auxiliary variable (SAV) approach and stabilized SAV approach in time and the Fourier-spectral method in space for the phase field crystal (PFC) equation. Unconditionally, energy stability is established for both first- and second-order fully discrete schemes. In addition to the stability, we also provide a rigorous error estimate which shows that our second-order in time with Fourier-spectral method in space converges with order Ot2 + Nm), where Δt, N, and m are time step size, number of Fourier modes in each direction, and regularity index in space, respectively. We also present numerical experiments to verify our theoretical results and demonstrate the robustness and accuracy of the schemes.

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References

  1. 1.

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

    MathSciNet  Article  Google Scholar 

  2. 2.

    Chen, C., Yang, X.: Fast, provably unconditionally energy stable, and second-order accurate algorithms for the anisotropic Cahn-Hilliard model. Comput. Methods Appl. Mech. Eng. 351, 35–59 (2019)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Elder, K., Grant, M.: Modeling elastic and plastic deformations in nonequilibrium processing using phase field crystals. Phys. Rev. E 70(5), 051605 (2004)

    Article  Google Scholar 

  4. 4.

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

    Article  Google Scholar 

  5. 5.

    Gomez, H., Nogueira, X.: An unconditionally energy-stable method for the phase field crystal equation. Comput. Methods Appl. Mech. Eng. 249, 52–61 (2012)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Guo, R., Xu, Y.: Local discontinuous Galerkin method and high order semi-implicit scheme for the phase field crystal equation. SIAM J. Sci. Comput. 38 (1), A105–A127 (2016)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Li, Q., Mei, L., Yang, X., Li, Y.: Efficient numerical schemes with unconditional energy stabilities for the modified phase field crystal equation. Adv. Comput. Math., 1–30 (2019)

  8. 8.

    Li, X., Shen, J., Rui, H.: Energy stability and convergence of SAV block-centered finite difference method for gradient flows. Math. Comput. 88(319), 2047–2068 (2019)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Li, Y., Kim, J.: An efficient and stable compact fourth-order finite difference scheme for the phase field crystal equation. Comput. Methods Appl. Mech. Eng. 319, 194–216 (2017)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Ramos, J.: C. canuto, my hussaini, a. quarteroni, ta zang, Spectral Methods in Fluid Dynamics. springer, New York (1991). dm 162

    Google Scholar 

  11. 11.

    Shen, J., Tang, T., Wang, L.L.: Spectral Methods: Algorithms, Analysis and Applications, vol. 41. Springer Science & Business Media (2011)

  12. 12.

    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)

    MathSciNet  Article  Google Scholar 

  13. 13.

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

    MathSciNet  Article  Google Scholar 

  14. 14.

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

    MathSciNet  Article  Google Scholar 

  15. 15.

    Swift, J., Hohenberg, P.C.: Hydrodynamic fluctuations at the convective instability. Phys. Rev. A 15(1), 319 (1977)

    Article  Google Scholar 

  16. 16.

    Wang, L.: A review of prolate spheroidal wave functions from the perspective of spectral methods. J. Math. Study 50(2), 101–143 (2017)

    MathSciNet  Article  Google Scholar 

  17. 17.

    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. 47(3), 2269–2288 (2009)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Yang, X.: Efficient schemes with unconditionally energy stability for the anisotropic Cahn-Hilliard equation using the stabilized-Scalar Augmented Variable (S-SAV) approach. arXiv:1804.02619 (2018)

  19. 19.

    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)

    MathSciNet  Article  Google Scholar 

  20. 20.

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

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgments

The authors express their thanks to the two referees for their helpful suggestions, which led to improvements of the presentation.

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Correspondence to Jie Shen.

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The work of X. Li is supported by the National Natural Science Foundation of China under grant number 11901489, 11971407 and Postdoctoral Science Foundation of China under grant numbers BX20190187 and 2019M650152. The work of J. Shen is supported in part by NSF grants DMS-1620262, DMS-1720442 and AFOSR grant FA9550-16-1-0102.

Communicated by: Carlos Garcia-Cervera

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Li, X., Shen, J. Stability and error estimates of the SAV Fourier-spectral method for the phase field crystal equation. Adv Comput Math 46, 48 (2020). https://doi.org/10.1007/s10444-020-09789-9

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Keywords

  • Phase field crystal
  • Fourier-spectral method
  • Scalar auxiliary variable (SAV)
  • Energy stability
  • Error estimates

Mathematics Subject Classification (2010)

  • 35G25
  • 65M12
  • 65M15
  • 65M70