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 O(Δt2 + N−m), 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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Ainsworth, M., Mao, Z.: Analysis and approximation of a fractional Cahn-Hilliard equation. SIAM J. Numer. Anal. 55(4), 1689–1718 (2017)
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)
Elder, K., Grant, M.: Modeling elastic and plastic deformations in nonequilibrium processing using phase field crystals. Phys. Rev. E 70(5), 051605 (2004)
Elder, K., Katakowski, M., Haataja, M., Grant, M.: Modeling elasticity in crystal growth. Phys. Rev. Lett. 88(24), 245701 (2002)
Gomez, H., Nogueira, X.: An unconditionally energy-stable method for the phase field crystal equation. Comput. Methods Appl. Mech. Eng. 249, 52–61 (2012)
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)
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)
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)
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)
Ramos, J.: C. canuto, my hussaini, a. quarteroni, ta zang, Spectral Methods in Fluid Dynamics. springer, New York (1991). dm 162
Shen, J., Tang, T., Wang, L.L.: Spectral Methods: Algorithms, Analysis and Applications, vol. 41. Springer Science & Business Media (2011)
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)
Shen, J., Xu, J., Yang, J.: The scalar auxiliary variable (SAV) approach for gradient flows. J. Comput. Phys. 353, 407–416 (2018)
Shen, J., Yang, X.: Numerical approximations of Allen-Cahn and Cahn-Hilliard equations. Discrete Contin. Dyn. Syst 28(4), 1669–1691 (2010)
Swift, J., Hohenberg, P.C.: Hydrodynamic fluctuations at the convective instability. Phys. Rev. A 15(1), 319 (1977)
Wang, L.: A review of prolate spheroidal wave functions from the perspective of spectral methods. J. Math. Study 50(2), 101–143 (2017)
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)
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)
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)
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)
The authors express their thanks to the two referees for their helpful suggestions, which led to improvements of the presentation.
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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
- Phase field crystal
- Fourier-spectral method
- Scalar auxiliary variable (SAV)
- Energy stability
- Error estimates
Mathematics Subject Classification (2010)