A Third Order Exponential Time Differencing Numerical Scheme for No-Slope-Selection Epitaxial Thin Film Model with Energy Stability

  • Kelong Cheng
  • Zhonghua Qiao
  • Cheng WangEmail author


In this paper we propose and analyze a (temporally) third order accurate exponential time differencing (ETD) numerical scheme for the no-slope-selection (NSS) equation of the epitaxial thin film growth model, with Fourier pseudo-spectral discretization in space. A linear splitting is applied to the physical model, and an ETD-based multistep approximation is used for time integration of the corresponding equation. In addition, a third order accurate Douglas-Dupont regularization term, in the form of \(-A {\Delta t}^2 \phi _0 (L_N) \Delta _N^2 ( u^{n+1} - u^n)\), is added in the numerical scheme. A careful Fourier eigenvalue analysis results in the energy stability in a modified version, and a theoretical justification of the coefficient A becomes available. As a result of this energy stability analysis, a uniform in time bound of the numerical energy is obtained. And also, the optimal rate convergence analysis and error estimate are derived in details, in the \(\ell ^\infty (0,T; H_h^1) \cap \ell ^2 (0,T; H_h^3)\) norm, with the help of a careful eigenvalue bound estimate, combined with the nonlinear analysis for the NSS model. This convergence estimate is the first such result for a third order accurate scheme for a gradient flow. Some numerical simulation results are presented to demonstrate the efficiency of the numerical scheme and the third order convergence. The long time simulation results for \(\varepsilon =0.02\) (up to \(T=3 \times 10^5\)) have indicated a logarithm law for the energy decay, as well as the power laws for growth of the surface roughness and the mound width. In particular, the power index for the surface roughness and the mound width growth, created by the third order numerical scheme, is more accurate than those produced by certain second order energy stable schemes in the existing literature.


Epitaxial thin film growth Slope selection Exponential time differencing Energy stability Optimal rate convergence analysis Aliasing error 

Mathematics Subject Classification

35K30 35K55 65L06 65M12 65M70 65T40 



This work is supported in part by the Longshan Talent Project of SWUST 18LZX529 (K. Cheng), Hong Kong Research Council GRF Grants 15300417 and 15325816, (Z. Qiao) and NSF DMS-1418689 (C. Wang).


  1. 1.
    Benesova, B., Melcher, C., Suli, E.: An implicit midpoint spectral approximation of nonlocal Cahn–Hilliard equations. Numer. Math. 52, 1466–1496 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Beylkin, G., Keiser, J.M., Vozovoi, L.: A new class of time discretization schemes for the solution of nonlinear PDEs. J. Comput. Phys. 147, 362–387 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Boyd, J.: Chebyshev and Fourier Spectral Methods. Dover, New York (2001)zbMATHGoogle Scholar
  4. 4.
    Canuto, C., Quarteroni, A.: Approximation results for orthogonal polynomials in Sobolev spaces. Math. Comput. 38, 67–86 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chen, W., Conde, S., Wang, C., Wang, X., Wise, S.M.: A linear energy stable scheme for a thin film model without slope selection. J. Sci. Comput. 52, 546–562 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chen, W., Li, W., Luo, Z., Wang, C., Wang, X.: A stabilized second order ETD multistep method for thin film growth model without slope selection. ESAIM Math. Model. Numer. Anal. (2019). Submitted and in review: arXiv:1907.02234Google Scholar
  7. 7.
    Chen, W., Wang, C., Wang, X., Wise, S.M.: A linear iteration algorithm for energy stable second order scheme for a thin film model without slope selection. J. Sci. Comput. 59, 574–601 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chen, W., Wang, X., Yan, Y., Zhang, Z.: A second order bdf numerical scheme with variable steps for the Cahn–Hilliard equation. SIAM J. Numer. Anal. 57(1), 495–525 (2019)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chen, W., Wang, Y.: A mixed finite element method for thin film epitaxy. Numer. Math. 122, 771–793 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Cheng, K., Feng, W., Wang, C., Wise, S.M.: An energy stable fourth order finite difference scheme for the Cahn–Hilliard equation. J. Comput. Appl. Math. 362, 574–595 (2019)CrossRefGoogle Scholar
  11. 11.
    Church, J.M., Guo, Z., Jimack, P.K., Madzvamuse, A., Promislow, K., Wise, S.M., Yang, F.: High accuracy benchmark problems for Allen–Cahn and Cahn–Hilliard dynamics. Commun. Comput. Phys. 26, 947–972 (2019)CrossRefGoogle Scholar
  12. 12.
    Cox, S.M., Matthews, P.C.: Exponential time differencing for stiff systems. J. Comput. Phys. 176, 430–455 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Ehrlich, G., Hudda, F.G.: Atomic view of surface diffusion: tungsten on tungsten. J. Chem. Phys. 44, 1036–1099 (1966)Google Scholar
  14. 14.
    Eyre, D.J.: Unconditionally gradient stable time marching the Cahn–Hilliard equation. MRS Symp. Proc. 529, 39 (1998)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Golubović, L.: Interfacial coarsening in epitaxial growth models without slope selection. Phys. Rev. Lett 78, 90–93 (1997)CrossRefGoogle Scholar
  16. 16.
    Gottlieb, D., Orszag, S.A.: Numerical Analysis of Spectral Methods, Theory and Applications. SIAM, Philadelphia (1977)CrossRefzbMATHGoogle Scholar
  17. 17.
    Gottlieb, S., Tone, F., Wang, C., Wang, X., Wirosoetisno, D.: Long time stability of a classical efficient scheme for two dimensional Navier–Stokes equations. SIAM J. Numer. Anal. 50, 126–150 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Gottlieb, S., Wang, C.: Stability and convergence analysis of fully discrete Fourier collocation spectral method for 3-D viscous Burgers’ equation. J. Sci. Comput. 53, 102–128 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Hochbruck, M., Ostermann, A.: Exponential integrators. Acta Numer. 19, 209–286 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Hochbruck, M., Ostermann, A.: Exponential multistep methods of Adams-type. BIT Numer. Math. 51, 889–908 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Ju, L., Li, X., Qiao, Z., Zhang, H.: Energy stability and convergence of exponential time differencing schemes for the epitaxial growth model without slope selection. Math. Comput. 87, 1859–1885 (2018)CrossRefzbMATHGoogle Scholar
  22. 22.
    Ju, L., Liu, X., Leng, W.: Compact implicit integration factor methods for a family of semilinear fourth-order parabolic equations. Discrete Contin. Dyn. Syst. Ser. B 19, 1667–1687 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Ju, L., Zhang, J., Du, Q.: Fast and accurate algorithms for simulating coarsening dynamics of Cahn–Hilliard equations. Comput. Mat. Sci. 108, 272–282 (2015)CrossRefGoogle Scholar
  24. 24.
    Ju, L., Zhang, J., Zhu, L., Du, Q.: Fast explicit integration factor methods for semilinear parabolic equations. J. Sci. Comput. 62, 431–455 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Kohn, R.V.: Energy-driven pattern formation. In: Sanz-Sole, M., Soria, J., Varona, J.L., Verdera, J. (eds.) Proceedings of the International Congress of Mathematicians, vol. 1, pp. 359–384. European Mathematical Society Publishing House, Madrid (2007)Google Scholar
  26. 26.
    Kohn, R.V., Yan, X.: Upper bound on the coarsening rate for an epitaxial growth model. Commun. Pure Appl. Math. 56, 1549–1564 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Lee, S., Kim, J.: Effective time step analysis of a nonlinear convex splitting scheme for the Cahn–Hilliard equation. Commun. Comput. Phys. 25, 448–460 (2019)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Li, B.: High-order surface relaxation versus the Ehrlich–Schwoebel effect. Nonlinearity 19, 2581–2603 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Li, B., Liu, J.: Thin film epitaxy with or without slope selection. Eur. J. Appl. Math. 14, 713–743 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Li, B., Liu, J.: Epitaxial growth without slope selection: energetics, coarsening, and dynamic scaling. J. Nonlinear Sci. 14, 429–451 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Li, D., Qiao, Z.: On second order semi-implicit Fourier spectral methods for 2D Cahn–Hilliard equations. J. Sci. Comput. 70, 301–341 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    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, 1653–1681 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Li, W., Chen, W., Wang, C., Yan, Y., He, R.: A second order energy stable linear scheme for a thin film model without slope selection. J. Sci. Comput. 76(3), 1905–1937 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Moldovan, D., Golubovic, L.: Interfacial coarsening dynamics in epitaxial growth with slope selection. Phys. Rev. E 61(6), 6190 (2000)CrossRefGoogle Scholar
  35. 35.
    Qiao, Z., Sun, Z., Zhang, Z.: The stability and convergence of two linearized finite difference schemes for the nonlinear epitaxial growth model. Numer. Methods Partial Differ. Equ. 28, 1893–1915 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Qiao, Z., Sun, Z., Zhang, Z.: Stability and convergence of second-order schemes for the nonlinear epitaxial growth model without slope selection. Math. Comput. 84, 653–674 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Qiao, Z., Wang, C., Wise, S.M., Zhang, Z.: Error analysis of a finite difference scheme for the epitaxial thin film growth model with slope selection with an improved convergence constant. Int. J. Numer. Anal. Model. 14, 283–305 (2017)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Qiao, Z., Zhang, Z., Tang, T.: An adaptive time-stepping strategy for the molecular beam epitaxy models. SIAM J. Sci. Comput. 33, 1395–1414 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Schwoebel, R.L.: Step motion on crystal surfaces: II. J. Appl. Phys. 40, 614–618 (1969)CrossRefGoogle Scholar
  40. 40.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Wang, C., Wang, X., Wise, S.M.: Unconditionally stable schemes for equations of thin film epitaxy. Discrete Contin. Dyn. Syst. 28, 405–423 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Wang, X., Ju, L., Du, Q.: Efficient and stable exponential time differencing Runge-Kutta methods for phase field elastic bending energy models. J. Comput. Phys. 316, 21–38 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Weinan, E.: Convergence of spectral methods for the Burgers’ equation. SIAM J. Numer. Anal. 29, 1520–1541 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Weinan, E.: Convergence of Fourier methods for Navier–Stokes equations. SIAM J. Numer. Anal. 30, 650–674 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Xu, C., Tang, T.: Stability analysis of large time-stepping methods for epitaxial growth models. SIAM J. Numer. Anal. 44(4), 1759–1779 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Yang, X., Zhao, J., Wang, Q.: Numerical approximations for the molecular beam epitaxial growth model based on the invariant energy quadratization method. J. Comput. Phys. 333, 104–127 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Zhu, L., Ju, L., Zhao, W.: Fast high-order compact exponential time differencing Runge–Kutta methods for second-order semilinear parabolic equations. J. Sci. Comput. 67, 1043–1065 (2016)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of ScienceSouthwest University of Science and TechnologyMianyangPeople’s Republic of China
  2. 2.Department of Applied MathematicsThe Hong Kong Polytechnic UniversityKowloonHong Kong
  3. 3.Department of MathematicsThe University of MassachusettsNorth DartmouthUSA

Personalised recommendations