Free Access
Issue
ESAIM: M2AN
Volume 54, Number 3, May-June 2020
Page(s) 727 - 750
DOI https://doi.org/10.1051/m2an/2019054
Published online 01 April 2020
  1. R.A. Adams and J.J.F. Fournier, Sobolev spaces, 2nd ed. (Academic press, Singapore, 2003). [Google Scholar]
  2. S. Agmon, Lectures on elliptic boundary value problems In: Vol. 369 of American Mathematical Soc. (2010). [Google Scholar]
  3. B. Benesova, C. Melcher and E. Suli, An implicit midpoint spectral approximation of nonlocal Cahn-Hilliard equations. SIAM J. Numer. Anal. 52 (2014) 1466–1496. [Google Scholar]
  4. G. Beylkin, J.M. Keiser and L. Vozovoi, A new class of time discretization schemes for the solution of nonlinear PDEs. J. Comput. Phys. 147 (1998) 362–387. [Google Scholar]
  5. C. Canuto and A. Quarteroni, Approximation results for orthogonal polynomials in sobolev spaces, Math. Comput. 38 (1982) 67–86. [Google Scholar]
  6. C. Canuto, M.Y. Hussaini, A. Quarteroni and T.A. Zang, Spectral Methods: Fundamentals in Single Domains. Springer (2006). [CrossRef] [Google Scholar]
  7. C. Canuto, M.Y. Hussaini, A. Quarteroni and T.A. Zang, Spectral Methods: Evolution to Complex Geometries and Applications to Fluid Dynamics. Springer Science & Business Media (2007). [CrossRef] [Google Scholar]
  8. W. Chen and Y. Wang, A mixed finite element method for thin film epitaxy. Numer. Math. 122 (2012) 771–793. [Google Scholar]
  9. W. Chen, S. Conde, C. Wang, X. Wang and S.M. Wise, A linear energy stable scheme for a thin film model without slope selection. J. Sci. Comput. 52 (2012) 546–562. [Google Scholar]
  10. W. Chen, C. Wang, X. Wang and S.M. Wise, A linear iteration algorithm for a second-order energy stable scheme for a thin film model without slope selection. J. Sci. Comput. 59 (2014) 574–601. [Google Scholar]
  11. K. Cheng, W. Feng, C. Wang and S.M. Wise, An energy stable fourth order finite difference scheme for the Cahn-Hilliard equation. J. Comput. Appl. Math. 362 (2019) 574–595. [Google Scholar]
  12. K. Cheng, Z. Qiao and C. Wang, A third order exponential time differencing numerical scheme for no-slope-selection epitaxial thin film model with energy stability. J. Sci. Comput. 81 (2019) 154–185. [Google Scholar]
  13. S.M. Cox and P.C. Matthews, Exponential time differencing for stiff systems. J. Comput. Phys. 176 (2002) 430–455. [Google Scholar]
  14. G. Ehrlich and F.G. Hudda, Atomic view of surface self-diffusion: Tungsten on tungsten. J. Chem. Phys. 44 (1966) 1039–1049. [Google Scholar]
  15. D.J. Eyre, Unconditionally gradient stable time marching the Cahn–Hilliard equation. In, Vol. 529 of Symposia BB - Computational & Mathematical Models of Microstructural Evolution (1998) 39. [Google Scholar]
  16. W. Feng, C. Wang, S.M. Wise and Z. Zhang, A second-order energy stable backward differentiation formula method for the epitaxial thin film equation with slope selection. Numer. Methods Partial Differ. Equ. 34 (2018) 1975–2007. [Google Scholar]
  17. L. Golubović, Interfacial coarsening in epitaxial growth models without slope selection. Phys. Rev. Lett. 78 (1997) 90–93. [Google Scholar]
  18. D. Gottlieb and S.A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications. SIAM, Philadelphia 26 (1977). [CrossRef] [Google Scholar]
  19. S. Gottlieb and C. Wang, Stability and convergence analysis of fully discrete fourier collocation spectral method for 3-D viscous burgers’ equation. J. Sci. Comput. 53 (2012) 102–128. [Google Scholar]
  20. M. Hochbruck and A. Ostermann, Exponential integrators. Acta Numer. 19 (2010) 209–286. [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
  21. M. Hochbruck and A. Ostermann, Exponential multistep methods of Adams-type. BIT Numer. Math. 51 (2011) 889–908. [Google Scholar]
  22. L. Ju, X. Liu and W. Leng, Compact implicit integration factor methods for a family of semilinear fourth-order parabolic equations. Discrete Cont. Dyn-B. 19 (2014) 1667–1687. [Google Scholar]
  23. L. Ju, J. Zhang and Q. Du, Fast and accurate algorithms for simulating coarsening dynamics of Cahn-Hilliard equations. Comput. Mater. Sci. 108 (2015) 272–282. [Google Scholar]
  24. L. Ju, J. Zhang, L. Zhu and Q. Du, Fast explicit integration factor methods for semilinear parabolic equations. J. Sci. Comput. 62 (2015) 431–455. [Google Scholar]
  25. L. Ju, X. Li, Z. Qiao and H. Zhang, Energy stability and convergence of exponential time differencing schemes for the epitaxial growth model without slope selection. Math. Comput. 87 (2018) 1859–1885. [Google Scholar]
  26. R.V. Kohn and X. Yan, Upper bound on the coarsening rate for an epitaxial growth model. Commun. Pur. Appl. Math. 56 (2003) 1549–1564. [CrossRef] [Google Scholar]
  27. B. Li, High-order surface relaxation versus the Ehrlich-Schwoebel effect. Nonlinearity 19 (2006) 2581–2603. [Google Scholar]
  28. B. Li and J. Liu, Thin film epitaxy with or without slope selection. Eur. J. Appl. Math. 14 (2003) 713–743. [Google Scholar]
  29. B. Li and J. Liu, Epitaxial growth without slope selection: energetics, coarsening, and dynamic scaling. J. Nonlinear Sci. 14 (2004) 429–451. [Google Scholar]
  30. D. Li and Z. Qiao, On the stabilization size of semi-implicit Fourier-spectral methods for 3D Cahn-Hilliard equations. Commun. Math. Sci. 15 (2017) 1489–1506. [Google Scholar]
  31. D. Li, Z. Qiao and T. Tang, Characterizing the stabilization size for semi-implicit Fourier-spectral method to phase field equations. SIAM J. Numer. Anal. 54 (2016) 1653–1681. [Google Scholar]
  32. X. Li, Z. Qiao and H. Zhang, Convergence of a fast explicit operator splitting method for the epitaxial growth model with slope selection. SIAM J. Numer. Anal. 55 (2017) 265–285. [Google Scholar]
  33. W. Li, W. Chen, C. Wang, Y. Yan and R. He, A second order energy stable linear scheme for a thin film model without slope selection. J. Sci. Comput. 76 (2018) 1905–1937. [Google Scholar]
  34. D. Moldovan and L. Golubovic, Interfacial coarsening dynamics in epitaxial growth with slope selection. Phys. Rev. E 61 (2000) 6190–6214. [Google Scholar]
  35. L. Nirenberg, On elliptic partial differential equations. IL Principio Di Minimo E Sue Applicazioni Alle Equazioni Funzionali 13 (1959) 1–48. [Google Scholar]
  36. Z. Qiao, Z. Sun and Z. Zhang, Stability and convergence of second-order schemes for the nonlinear epitaxial growth model without slope selection. Math. Comput. 84 (2015) 653–674. [Google Scholar]
  37. Z. Qiao, T. Tang and H. Xie, Error analysis of a mixed finite element method for the molecular beam epitaxy model. SIAM J. Numer. Anal. 53 (2015) 184–205. [Google Scholar]
  38. Z. Qiao, C. Wang, S.M. Wise and Z. Zhang, Error analysis of a finite difference scheme for the epitaxial thin film model with slope selection with an improved convergence constant. Int. J. Numer. Anal. Mod. 14 (2017) 283–305. [Google Scholar]
  39. R.L. Schwoebel, Step motion on crystal surfaces II. J. Appl. Phys. 40 (1969) 614–618. [Google Scholar]
  40. J. Shen, T. Tang and L. Wang, Spectral Methods: Algorithms, Analysis and Applications. Springer Science & Business Media 41 (2011). [Google Scholar]
  41. J. Shen, C. Wang, X. Wang and S.M. Wise, Second-order convex splitting schemes for gradient flows with Ehrlich-Schwoebel type energy: application to thin film epitaxy. SIAM J. Numer. Anal. 50 (2012) 105–125. [Google Scholar]
  42. J. Shen, J. Xu and J. Yang, The scalar auxiliary variable (sav) approach for gradient flows. J. Comput. Phys. 353 (2018) 407–416. [Google Scholar]
  43. C. Wang, X. Wang and S.M. Wise, Unconditionally stable schemes for equations of thin film epitaxy. Discrete Contin. Dyn. Syst. 28 (2010) 405–423. [CrossRef] [Google Scholar]
  44. X. Wang, L. Ju and Q. Du, Efficient and stable exponential time differencing Runge-Kutta methods for phase field elastic bending energy models. J. Comput. Phys. 316 (2016) 21–38. [Google Scholar]
  45. C. Xu and T. Tang, Stability analysis of large time-stepping methods for epitaxial growth models. SIAM J. Numer. Anal. 44 (2006) 1759–1779. [Google Scholar]
  46. X. Yang, J. Zhao and Q. Wang, Numerical approximations for the molecular beam epitaxial growth model based on the invariant energy quadratization method. J. Comput. Phys. 333 (2017) 104–127. [Google Scholar]
  47. L. Zhu, L. Ju and W. Zhao, Fast high-order compact exponential time differencing Runge-Kutta methods for second-order semilinear parabolic equations. J. Sci. Comput. 67 (2016) 1043–1065. [Google Scholar]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.

Recommended for you