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The BDF3/EP3 Scheme for MBE with No Slope Selection is Stable

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A Correction to this article was published on 01 November 2021

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Abstract

We consider the classical molecular beam epitaxy (MBE) model with logarithmic type potential known as no-slope-selection. We employ a third order backward differentiation (BDF3) in time with implicit treatment of the surface diffusion term. The nonlinear term is approximated by a third order explicit extrapolation (EP3) formula. We exhibit mild time step constraints under which the modified energy dissipation law holds. We break the second Dahlquist barrier and develop a new theoretical framework to prove unconditional uniform energy boundedness with no size restrictions on the time step. This is the first unconditional result for third order BDF methods applied to the MBE models without introducing any stabilization term or fictitious variable. The analysis can be generalized to a restrictive class of phase field models whose nonlinearity has bounded derivatives. A novel theoretical framework is also established for the error analysis of high order methods.

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References

  1. Chen, W., Conde, S., Wang, C., Wang, X., Wise, S.: A linear energy stable scheme for a thin film model without slope selection. J. Sci. Comput. 52(3), 546–562 (2012)

    Article  MathSciNet  Google Scholar 

  2. Creedon, D.M., Miller, J.: The stability properties of \(q\)-step backward difference schemes. BIT Numer. Math. 15(3), 244–249 (1975)

    Article  MathSciNet  Google Scholar 

  3. Dahlquist, G.: A special stability problem for linear multistep methods. BIT 3, 27–33 (1963)

    Article  MathSciNet  Google Scholar 

  4. Cryer, C.W.: On the instability of high order backward-difference multistep methods. BIT Numer. Math. 12(1), 17–25 (1972)

    Article  MathSciNet  Google Scholar 

  5. Endre, S., Mayers, D.: An Introduction to Numerical Analysis. Cambridge University Press, Cambridge (2003).. (ISBN 0521007941)

    MATH  Google Scholar 

  6. Eyre, D.J.: Unconditionally gradient stable time marching the Cahn–Hilliard equation. MRS Online Proc. Libr. Arch. 529, 1998 (1998)

    MathSciNet  Google Scholar 

  7. Fredebeul, C.: A-BDF: a generalization of the backward differentiation formulae. SIAM J. Numer. Anal. 35(5), 1917–1938 (1998)

    Article  MathSciNet  Google Scholar 

  8. Hao, Y., Huang, Q., Wang, C.: A third order BDF energy stable linear scheme for the no-slope-selection thin film model

  9. Herring, C.: Surface tension as a motivation for sintering. In: Kingston, W.E. (ed.) The Physics of powder Metallurgy. McGraw-Hill, New York (1951)

    Google Scholar 

  10. Iserles, A.: A First Course in the Numerical Analysis of Differential Equations. Cambridge University Press, Cambridge (1996) ISBN 978-0-521-55655-2

  11. Li, B., Liu, J.G.: Thin film epitaxy with or without slope selection. Eur. J. Appl. Math. 14(6), 713–743 (2003)

    Article  MathSciNet  Google Scholar 

  12. Li, D., Qiao, Z., Tang, T.: Characterizing the stabilization size for semi-implicit Fourier-spectral method to phase field equations. SIAM J. Numer. Anal. 54(3), 1653–1681 (2016)

    Article  MathSciNet  Google Scholar 

  13. Li, D., Qiao, Z., Tang, T.: Gradient bounds for a thin film epitaxy equation. J. Differ. Equ. 262(3), 1720–1746 (2017)

    Article  MathSciNet  Google Scholar 

  14. Li, D., Qiao, Z.: On second order semi-implicit Fourier spectral methods for 2D Cahn–Hilliard equations. J. Sci. Comput. 70, 301–341 (2017)

    Article  MathSciNet  Google Scholar 

  15. Li, D., Qiao, Z.: On the stabilization size of semi-implicit Fourier-spectral methods for 3D Cahn–Hilliard equations. Commun. Math. Sci. 15(6), 1489–1506 (2017)

    Article  MathSciNet  Google Scholar 

  16. Li, D., Wang, F., Yang, K.: An improved gradient bound for 2D MBE. J. Differ. Equ. 269(12), 11165–11171 (2020)

    Article  MathSciNet  Google Scholar 

  17. Li, D., Tang, T.: Stability of the semi-implicit method for the Cahn–Hilliard equation with logarithmic potentials. Ann. Appl. Math. 37, 31–60 (2021)

    Article  MathSciNet  Google Scholar 

  18. Li, D., Quan, C., Tang, T.: Stability and convergence analysis for the implicit-explicit method to the Cahn-Hilliard equation. Math. Comp. (to appear)

  19. Li, D.: Effective maximum principles for spectral methods. Ann. Appl. Math. 37, 131–290 (2021)

    Article  MathSciNet  Google Scholar 

  20. Mullins, W.W.: Theory of thermal grooving. J. Appl. Phys. 28(3), 333–339 (1957)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  22. Shen, J., Wang, C., Wang, X., Wise, S.M.: Second-order convex splitting schemes for gradient flows with Ehrlich-Schwoebel type energy: application to thin film epitaxy. SIAM J. Numer. Anal. 50, 105–125 (2012)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  24. 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  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  26. Song, H., Shu, C.W.: Unconditional energy stability analysis of a second order implicit–explicit local discontinuous Galerkin method for the Cahn–Hilliard equation. J. Sci. Comput. 73, 1178–1203 (2017)

    Article  MathSciNet  Google Scholar 

  27. Wang, C., Wang, X.M., Wise, S.M.: Unconditionally stable schemes for equations of thin film epitaxy. Discrete Contin. Dyn. Syst. A 28(1), 405 (2010)

    Article  MathSciNet  Google Scholar 

  28. Xu, C.J., Tang, T.: Stability analysis of large time-stepping methods for epitaxial growth models. SIAM J. Numer. Anal. 44(4), 1759–1779 (2006)

    Article  MathSciNet  Google Scholar 

  29. Zhu, J.Z., Chen, L.-Q., Shen, J., Tikare, V.: Coarsening kinetics from a variable-mobility Cahn–Hilliard equation: application of a semi-implicit Fourier spectral method. Phys. Rev. E 60(4), 3564 (1999)

    Article  Google Scholar 

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Acknowledgements

The research of W. Yang is supported by NSFC Grants 11801550 and 11871470. The work of C. Quan is supported by NSFC Grant 11901281, the Guangdong Basic and Applied Basic Research Foundation (2020A1515010336), and the Stable Support Plan Program of Shenzhen Natural Science Fund (Program Contract No. 20200925160747003).

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Authors and Affiliations

  1. SUSTech International Center for Mathematics and Department of Mathematics, Southern University of Science and Technology, Shenzhen, 518055, People’s Republic of China

    Dong Li

  2. SUSTech International Center for Mathematics, Southern University of Science and Technology, Shenzhen, 518055, People’s Republic of China

    Chaoyu Quan

  3. Wuhan Institute of Physics and Mathematics, Innovation Academy for Precision Measurement Science and Technology, Chinese Academy of Sciences, Wuhan, 430071, People’s Republic of China

    Wen Yang

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  1. Dong Li
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  2. Chaoyu Quan
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  3. Wen Yang
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Correspondence to Dong Li.

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Li, D., Quan, C. & Yang, W. The BDF3/EP3 Scheme for MBE with No Slope Selection is Stable. J Sci Comput 89, 33 (2021). https://doi.org/10.1007/s10915-021-01642-5

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