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A Third Order Exponential Time Differencing Numerical Scheme for No-Slope-Selection Epitaxial Thin Film Model with Energy Stability

  • Kelong Cheng
  • Zhonghua Qiao
  • Cheng WangEmail author
Article
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Abstract

In this paper we propose and analyze a (temporally) third order accurate exponential time differencing (ETD) numerical scheme for the no-slope-selection (NSS) equation of the epitaxial thin film growth model, with Fourier pseudo-spectral discretization in space. A linear splitting is applied to the physical model, and an ETD-based multistep approximation is used for time integration of the corresponding equation. In addition, a third order accurate Douglas-Dupont regularization term, in the form of \(-A {\Delta t}^2 \phi _0 (L_N) \Delta _N^2 ( u^{n+1} - u^n)\), is added in the numerical scheme. A careful Fourier eigenvalue analysis results in the energy stability in a modified version, and a theoretical justification of the coefficient A becomes available. As a result of this energy stability analysis, a uniform in time bound of the numerical energy is obtained. And also, the optimal rate convergence analysis and error estimate are derived in details, in the \(\ell ^\infty (0,T; H_h^1) \cap \ell ^2 (0,T; H_h^3)\) norm, with the help of a careful eigenvalue bound estimate, combined with the nonlinear analysis for the NSS model. This convergence estimate is the first such result for a third order accurate scheme for a gradient flow. Some numerical simulation results are presented to demonstrate the efficiency of the numerical scheme and the third order convergence. The long time simulation results for \(\varepsilon =0.02\) (up to \(T=3 \times 10^5\)) have indicated a logarithm law for the energy decay, as well as the power laws for growth of the surface roughness and the mound width. In particular, the power index for the surface roughness and the mound width growth, created by the third order numerical scheme, is more accurate than those produced by certain second order energy stable schemes in the existing literature.

Keywords

Epitaxial thin film growth Slope selection Exponential time differencing Energy stability Optimal rate convergence analysis Aliasing error 

Mathematics Subject Classification

35K30 35K55 65L06 65M12 65M70 65T40 

Notes

Acknowledgements

This work is supported in part by the Longshan Talent Project of SWUST 18LZX529 (K. Cheng), Hong Kong Research Council GRF Grants 15300417 and 15325816, (Z. Qiao) and NSF DMS-1418689 (C. Wang).

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of ScienceSouthwest University of Science and TechnologyMianyangPeople’s Republic of China
  2. 2.Department of Applied MathematicsThe Hong Kong Polytechnic UniversityKowloonHong Kong
  3. 3.Department of MathematicsThe University of MassachusettsNorth DartmouthUSA

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