A stabilized second order exponential time differencing multistep method for thin film growth model without slope selection

Chen, Wenbin; Li, Weijia; Luo, Zhiwen; Wang, Cheng; Wang, Xiaoming

doi:10.1051/m2an/2019054

Chen, Wenbin ; Li, Weijia ; Luo, Zhiwen ; Wang, Cheng ; Wang, Xiaoming

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 54 (2020) no. 3, pp. 727-750.

Suite au passage du modèle économique de la revue en S20, le texte intégral des articles des années 2019 et 2020 est accessible uniquement sur le site de l'éditeur et est réservé aux abonnés.

Résumé

In this paper, a stabilized second order in time accurate linear exponential time differencing (ETD) scheme for the no-slope-selection thin film growth model is presented. An artificial stabilizing term $A τ^{2} \frac{\partial Δ^{2} u}{\partial t}$ is added to the physical model to achieve energy stability, with ETD-based multi-step approximations and Fourier collocation spectral method applied in the time integral and spatial discretization of the evolution equation, respectively. Long time energy stability and detailed $𝓁^{\infty} (0, T; 𝓁^{2})$ error analysis are provided based on the energy method, with a careful estimate of the aliasing error. In addition, numerical experiments are presented to demonstrate the energy decay and convergence rate.

MR Zbl

DOI : 10.1051/m2an/2019054

Classification : 65M12, 65M70, 65Z05
Mots-clés : Epitaxial thin film growth, exponential time differencing, long time energy stability, convergence analysis, second order scheme

@article{M2AN_2020__54_3_727_0,
     author = {Chen, Wenbin and Li, Weijia and Luo, Zhiwen and Wang, Cheng and Wang, Xiaoming},
     title = {A stabilized second order exponential time differencing multistep method for thin film growth model without slope selection},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {727--750},
     publisher = {EDP-Sciences},
     volume = {54},
     number = {3},
     year = {2020},
     doi = {10.1051/m2an/2019054},
     mrnumber = {4080786},
     zbl = {1437.65149},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an/2019054/}
}

TY  - JOUR
AU  - Chen, Wenbin
AU  - Li, Weijia
AU  - Luo, Zhiwen
AU  - Wang, Cheng
AU  - Wang, Xiaoming
TI  - A stabilized second order exponential time differencing multistep method for thin film growth model without slope selection
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2020
SP  - 727
EP  - 750
VL  - 54
IS  - 3
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an/2019054/
DO  - 10.1051/m2an/2019054
LA  - en
ID  - M2AN_2020__54_3_727_0
ER  -

%0 Journal Article
%A Chen, Wenbin
%A Li, Weijia
%A Luo, Zhiwen
%A Wang, Cheng
%A Wang, Xiaoming
%T A stabilized second order exponential time differencing multistep method for thin film growth model without slope selection
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2020
%P 727-750
%V 54
%N 3
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an/2019054/
%R 10.1051/m2an/2019054
%G en
%F M2AN_2020__54_3_727_0

Chen, Wenbin; Li, Weijia; Luo, Zhiwen; Wang, Cheng; Wang, Xiaoming. A stabilized second order exponential time differencing multistep method for thin film growth model without slope selection. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 54 (2020) no. 3, pp. 727-750. doi : 10.1051/m2an/2019054. http://www.numdam.org/articles/10.1051/m2an/2019054/

Bibliographie
Cité par

R.A. Adams and J.J.F. Fournier, Sobolev spaces, 2nd ed. (Academic press, Singapore, 2003). | MR | Zbl

S. Agmon, Lectures on elliptic boundary value problems In: Vol. 369 of American Mathematical Soc. (2010). | MR | Zbl

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. | DOI | MR

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. | DOI | MR | Zbl

C. Canuto and A. Quarteroni, Approximation results for orthogonal polynomials in sobolev spaces, Math. Comput. 38 (1982) 67–86. | DOI | MR | Zbl

C. Canuto, M.Y. Hussaini, A. Quarteroni and T.A. Zang, Spectral Methods: Fundamentals in Single Domains. Springer (2006). | DOI | MR

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). | DOI | MR | Zbl

W. Chen and Y. Wang, A mixed finite element method for thin film epitaxy. Numer. Math. 122 (2012) 771–793. | DOI | MR | Zbl

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. | DOI | MR | Zbl

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. | DOI | MR | Zbl

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. | DOI | MR

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. | DOI | MR

S.M. Cox and P.C. Matthews, Exponential time differencing for stiff systems. J. Comput. Phys. 176 (2002) 430–455. | DOI | MR | Zbl

G. Ehrlich and F.G. Hudda, Atomic view of surface self-diffusion: Tungsten on tungsten. J. Chem. Phys. 44 (1966) 1039–1049. | DOI

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. | MR

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. | DOI | MR

L. Golubović, Interfacial coarsening in epitaxial growth models without slope selection. Phys. Rev. Lett. 78 (1997) 90–93. | DOI

D. Gottlieb and S.A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications. SIAM, Philadelphia 26 (1977). | MR | Zbl

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. | DOI | MR | Zbl

M. Hochbruck and A. Ostermann, Exponential integrators. Acta Numer. 19 (2010) 209–286. | DOI | MR | Zbl

M. Hochbruck and A. Ostermann, Exponential multistep methods of Adams-type. BIT Numer. Math. 51 (2011) 889–908. | DOI | MR | Zbl

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. | MR | Zbl

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. | DOI

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. | DOI | MR

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. | DOI | MR

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. | DOI | MR | Zbl

B. Li, High-order surface relaxation versus the Ehrlich-Schwoebel effect. Nonlinearity 19 (2006) 2581–2603. | DOI | MR | Zbl

B. Li and J. Liu, Thin film epitaxy with or without slope selection. Eur. J. Appl. Math. 14 (2003) 713–743. | DOI | MR | Zbl

B. Li and J. Liu, Epitaxial growth without slope selection: energetics, coarsening, and dynamic scaling. J. Nonlinear Sci. 14 (2004) 429–451. | DOI | MR | Zbl

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. | DOI | MR

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. | DOI | MR

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. | DOI | MR

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. | DOI | MR

D. Moldovan and L. Golubovic, Interfacial coarsening dynamics in epitaxial growth with slope selection. Phys. Rev. E 61 (2000) 6190–6214. | DOI

L. Nirenberg, On elliptic partial differential equations. IL Principio Di Minimo E Sue Applicazioni Alle Equazioni Funzionali 13 (1959) 1–48. | Zbl

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. | DOI | MR | Zbl

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. | DOI | MR

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. | MR

R.L. Schwoebel, Step motion on crystal surfaces II. J. Appl. Phys. 40 (1969) 614–618. | DOI

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

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. | DOI | MR | Zbl

J. Shen, J. Xu and J. Yang, The scalar auxiliary variable (sav) approach for gradient flows. J. Comput. Phys. 353 (2018) 407–416. | DOI | MR

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. | DOI | MR | Zbl

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. | DOI | MR

C. Xu and T. Tang, Stability analysis of large time-stepping methods for epitaxial growth models. SIAM J. Numer. Anal. 44 (2006) 1759–1779. | DOI | MR | Zbl

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. | DOI | MR

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. | DOI | MR

Cité par Sources :