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

Kelong Cheng, Zhonghua Qiao, Cheng Wang

Research output: Journal article publicationJournal articleAcademic researchpeer-review

90 Citations (Scopus)


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 -AΔt2ϕ0(LN)ΔN2(un+1-un), 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 ℓ∞(0,T;Hh1)∩ℓ2(0,T;Hh3) 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 ε= 0.02 (up to T= 3 × 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.

Original languageEnglish
Pages (from-to)154-185
Number of pages32
JournalJournal of Scientific Computing
Issue number1
Publication statusPublished - 1 Oct 2019


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

ASJC Scopus subject areas

  • Software
  • Theoretical Computer Science
  • Numerical Analysis
  • General Engineering
  • Computational Theory and Mathematics
  • Computational Mathematics
  • Applied Mathematics


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