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Stability and convergence of second-order schemes for the nonlinear epitaxial growth model without slope selection


Authors: Zhonghua Qiao, Zhi-zhong Sun and Zhengru Zhang
Journal: Math. Comp. 84 (2015), 653-674
MSC (2010): Primary 65M06, 65M12, 65Z05
DOI: https://doi.org/10.1090/S0025-5718-2014-02874-3
Published electronically: July 17, 2014
MathSciNet review: 3290959
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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.


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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

DOI: https://doi.org/10.1090/S0025-5718-2014-02874-3
Keywords: Molecular beam epitaxy, finite difference scheme, energy decay, stability, convergence, linearized difference scheme
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
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.