Stability and convergence of second-order schemes for the nonlinear epitaxial growth model without slope selection
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Abstract:
We present one nonlinear and one linearized numerical schemes for the nonlinear epitaxial growth model without slope selection. Both schemes are proved to be uniquely solvable and convergent with the convergence rate of order two in a discrete $L_2$-norm. By introducing an auxiliary variable in the discrete energy functional, the energy stability of both schemes is guaranteed regardless of the time step size, in the sense that a modified energy is monotonically nonincreasing in discrete time. Numerical experiments are carried out to support the theoretical claims.References
- Georgios D. Akrivis, Finite difference discretization of the cubic Schrödinger equation, IMA J. Numer. Anal. 13 (1993), no. 1, 115–124. MR 1199033, DOI 10.1093/imanum/13.1.115
- Wenbin Chen, Sidafa Conde, Cheng Wang, Xiaoming Wang, and Steven M. Wise, A linear energy stable scheme for a thin film model without slope selection, J. Sci. Comput. 52 (2012), no. 3, 546–562. MR 2948706, DOI 10.1007/s10915-011-9559-2
- Wenbin Chen, Cheng Wang, Xiaoming Wang, and Steven 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), no. 3, 574–601. MR 3199497, DOI 10.1007/s10915-013-9774-0
- Qiang Du and R. A. Nicolaides, Numerical analysis of a continuum model of phase transition, SIAM J. Numer. Anal. 28 (1991), no. 5, 1310–1322. MR 1119272, DOI 10.1137/0728069
- J.W. Evans and P.A. Thiel, A little chemistry helps the big get Bigger, Science, 330 (2010), 599-600.
- J.W. Evans, P.A. Thiel and M.C. Bartelt, Morphological evolution during epitaxial thin film growth: formation of 2D islands and 3D mounds, Surf. Sci. Rep., 61 (2006), 1-128.
- David J. Eyre, Unconditionally gradient stable time marching the Cahn-Hilliard equation, Computational and mathematical models of microstructural evolution (San Francisco, CA, 1998) Mater. Res. Soc. Sympos. Proc., vol. 529, MRS, Warrendale, PA, 1998, pp. 39–46. MR 1676409, DOI 10.1557/PROC-529-39
- Daisuke Furihata, A stable and conservative finite difference scheme for the Cahn-Hilliard equation, Numer. Math. 87 (2001), no. 4, 675–699. MR 1815731, DOI 10.1007/PL00005429
- L. Golubović, Interfacial coarsening in epitaxial growth models without slope selection, Phys. Rev. Lett., 78 (1997), 90-93.
- F. Guillén-González and G. Tierra, On linear schemes for a Cahn-Hilliard diffuse interface model, J. Comput. Phys. 234 (2013), 140–171. MR 2999772, DOI 10.1016/j.jcp.2012.09.020
- Yinnian He, Yunxian Liu, and Tao Tang, On large time-stepping methods for the Cahn-Hilliard equation, Appl. Numer. Math. 57 (2007), no. 5-7, 616–628. MR 2322435, DOI 10.1016/j.apnum.2006.07.026
- Z. Hu, S. M. Wise, C. Wang, and J. S. Lowengrub, Stable and efficient finite-difference nonlinear-multigrid schemes for the phase field crystal equation, J. Comput. Phys. 228 (2009), no. 15, 5323–5339. MR 2541456, DOI 10.1016/j.jcp.2009.04.020
- M.D. Johnson, C. Orme, A.W. Hunt, D. Graff, J. Sudijono, L.M. Sander and B.G. Orr, Stable and unstable growth in molecular beam epitaxy, Phys. Rev. Lett., 72 (1994), 116-119.
- Robert V. Kohn and Xiaodong Yan, Upper bound on the coarsening rate for an epitaxial growth model, Comm. Pure Appl. Math. 56 (2003), no. 11, 1549–1564. MR 1995869, DOI 10.1002/cpa.10103
- Bo Li and Jian-Guo Liu, Thin film epitaxy with or without slope selection, European J. Appl. Math. 14 (2003), no. 6, 713–743. MR 2034852, DOI 10.1017/S095679250300528X
- Bo Li and Jian-Guo Liu, Epitaxial growth without slope selection: energetics, coarsening, and dynamic scaling, J. Nonlinear Sci. 14 (2004), no. 5, 429–451 (2005). MR 2126166, DOI 10.1007/s00332-004-0634-9
- Juan Li, ZhiZhong Sun, and Xuan Zhao, A three level linearized compact difference scheme for the Cahn-Hilliard equation, Sci. China Math. 55 (2012), no. 4, 805–826. MR 2903464, DOI 10.1007/s11425-011-4290-x
- Zhonghua Qiao, Zhi-zhong Sun, and Zhengru Zhang, The stability and convergence of two linearized finite difference schemes for the nonlinear epitaxial growth model, Numer. Methods Partial Differential Equations 28 (2012), no. 6, 1893–1915. MR 2981875, DOI 10.1002/num.20707
- Zhonghua Qiao, Zhengru Zhang, and Tao Tang, An adaptive time-stepping strategy for the molecular beam epitaxy models, SIAM J. Sci. Comput. 33 (2011), no. 3, 1395–1414. MR 2813245, DOI 10.1137/100812781
- L. Ratke and P. W. Voorhees, Growth and coarsening, Springer-Verlag, Berlin, 2002.
- A. A. Samarskiĭ and V. B. Andreev, Raznostnye metody dlya èllipticheskikh uravneniĭ, Izdat. “Nauka”, Moscow, 1976 (Russian). MR 0502017
- Jie Shen, Cheng Wang, Xiaoming Wang, and Steven 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), no. 1, 105–125. MR 2888306, DOI 10.1137/110822839
- Zhi Zhong Sun, A second-order accurate linearized difference scheme for the two-dimensional Cahn-Hilliard equation, Math. Comp. 64 (1995), no. 212, 1463–1471. MR 1308465, DOI 10.1090/S0025-5718-1995-1308465-4
- Zhi-Zhong Sun and Qi-Ding Zhu, On Tsertsvadze’s difference scheme for the Kuramoto-Tsuzuki equation, J. Comput. Appl. Math. 98 (1998), no. 2, 289–304. MR 1657022, DOI 10.1016/S0377-0427(98)00135-6
- Cheng Wang, Xiaoming Wang, and Steven M. Wise, Unconditionally stable schemes for equations of thin film epitaxy, Discrete Contin. Dyn. Syst. 28 (2010), no. 1, 405–423. MR 2629487, DOI 10.3934/dcds.2010.28.405
- S. M. Wise, C. Wang, and J. S. Lowengrub, An energy-stable and convergent finite-difference scheme for the phase field crystal equation, SIAM J. Numer. Anal. 47 (2009), no. 3, 2269–2288. MR 2519603, DOI 10.1137/080738143
- Chuanju Xu and Tao Tang, Stability analysis of large time-stepping methods for epitaxial growth models, SIAM J. Numer. Anal. 44 (2006), no. 4, 1759–1779. MR 2257126, DOI 10.1137/050628143
- Zhengru Zhang and Zhonghua Qiao, An adaptive time-stepping strategy for the Cahn-Hilliard equation, Commun. Comput. Phys. 11 (2012), no. 4, 1261–1278. MR 2864085, DOI 10.4208/cicp.300810.140411s
- Yu Lin Zhou, Applications of discrete functional analysis to the finite difference method, International Academic Publishers, Beijing, 1991. MR 1133399
Additional Information
- Zhonghua Qiao
- Affiliation: Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong.
- Email: zqiao@polyu.edu.hk
- Zhi-zhong Sun
- Affiliation: Department of Mathematics, Southeast University, Nanjing, 210096, People’s Republic of China.
- Email: zzsun@seu.edu.cn
- Zhengru Zhang
- Affiliation: Laboratory of Mathematics and Complex Systems, Ministry of Education and School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, People’s Republic of China.
- Email: zrzhang@bnu.edu.cn
- Received by editor(s): October 13, 2012
- Received by editor(s) in revised form: June 4, 2013
- Published electronically: July 17, 2014
- Additional Notes: The research of the first author was partially supported by the Hong Kong RGC grant PolyU 2021/12P and the Hong Kong Polytechnic University grants A-PL61 and 1-ZV9Y
The second author was supported by NSFC under Grant 11271068
The research of the third author was supported by NSFC under Grants 11071124, 11271048, 91130021 and the Fundamental Research Funds for the Central Universities - © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp. 84 (2015), 653-674
- MSC (2010): Primary 65M06, 65M12, 65Z05
- DOI: https://doi.org/10.1090/S0025-5718-2014-02874-3
- MathSciNet review: 3290959