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Energy stability and error estimates of exponential time differencing schemes for the epitaxial growth model without slope selection


Authors: Lili Ju, Xiao Li, Zhonghua Qiao and Hui Zhang
Journal: Math. Comp. 87 (2018), 1859-1885
MSC (2010): Primary 35Q99, 47A56, 65M12, 65M70
DOI: https://doi.org/10.1090/mcom/3262
Published electronically: September 21, 2017
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Abstract: In this paper, we propose a class of exponential time differencing (ETD) schemes for solving the epitaxial growth model without slope selection. A linear convex splitting is first applied to the energy functional of the model, and then Fourier collocation and ETD-based multistep approximations are used respectively for spatial discretization and time integration of the corresponding gradient flow equation. Energy stabilities and error estimates of the first and second order ETD schemes are rigorously established in the fully discrete sense. We also numerically demonstrate the accuracy of the proposed schemes and simulate the coarsening dynamics with small diffusion coefficients. The results show the logarithm law for the energy decay and the power laws for growth of the surface roughness and the mound width, which are consistent with the existing theories in the literature.


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Additional Information

Lili Ju
Affiliation: Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208
Email: ju@math.sc.edu

Xiao Li
Affiliation: Applied and Computational Mathematics Division, Beijing Computational Science Research Center, Beijing, 100193, People’s Republic of China
Email: xiaoli@csrc.ac.cn

Zhonghua Qiao
Affiliation: Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong
Email: zhonghua.qiao@polyu.edu.hk

Hui 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: hzhang@bnu.edu.cn

DOI: https://doi.org/10.1090/mcom/3262
Keywords: Thin film growth, exponential time differencing, Fourier collocation, linear convex splitting, energy stability, error estimates
Received by editor(s): June 10, 2016
Received by editor(s) in revised form: November 7, 2016, and January 4, 2017
Published electronically: September 21, 2017
Additional Notes: The first author’s research was partially supported by the US National Science Foundation grant DMS-1521965.
The second author’s research was partially supported by the Hong Kong Research Grant Council GRF grant 15302214 during his visit at the Hong Kong Polytechnic University and China Postdoctoral Science Foundation grant 2017M610748.
The third author is the corresponding author. The third author’s research was partially supported by the Hong Kong Research Grant Council GRF grant 15302214, NSFC/RGC Joint Research Scheme N_HKBU204/12 and the Hong Kong Polytechnic University internal grant 1-ZE33.
The fourth author’s research was partially supported by NSFC/RGC Joint Research Scheme 11261160486, NSFC grants 11471046, 11571045 and the Ministry of Education Program for New Century Excellent Talents Project NCET-12-0053.
Article copyright: © Copyright 2017 American Mathematical Society

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