Artificial Regularization Parameter Analysis for the No-Slope-Selection Epitaxial Thin Film Model

Xiangjun Meng, Zhonghua Qiao, Cheng Wang, Zhengru Zhang

Research output: Journal article publicationJournal articleAcademic researchpeer-review

24 Citations (Scopus)

Abstract

In this paper we study the effect of the artificial regularization term for the second order accurate (in time) numerical schemes for the no-slope-selection (NSS) equation of the epitaxial thin film growth model. In particular, we propose and analyze an alternate second order backward differentiation formula (BDF) scheme, with Fourier pseudo-spectral spatial discretization. The surface diffusion term is treated implicitly, while the nonlinear chemical potential is approximated by a second order explicit extrapolation formula. A second order accurate Douglas-Dupont regularization term, in the form of −A∆t∆2N(un+1−un), is added in the numerical scheme to justify the energy stability at a theoretical level. Due to an alternate expression of the nonlinear chemical potential terms, such a numerical scheme requires a minimum value of the artificial regularization parameter as A=1024289 , much smaller than the other reported artificial parameter values in the existing literature. Such an optimization of the artificial parameter value is expected to reduce the numerical diffusion, and henceforth improve the long time numerical accuracy. Moreover, the optimal rate convergence analysis and error estimate are derived in details, in the ℓ(0,T;ℓ2)∩ℓ2(0,T;Hh2) norm, with the help of a linearized estimate for the nonlinear error terms. Some numerical simulation results are presented to demonstrate the efficiency and accuracy of the alternate second order numerical scheme. The long time simulation results for ε=0.02 (up to T=3×105) 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.

Original languageEnglish
Pages (from-to)441-462
Number of pages22
JournalCSIAM Transactions on Applied Mathematics
Volume1
Issue number3
DOIs
Publication statusPublished - 1 Sept 2020

Keywords

  • Douglas-Dupont regularization
  • energy stability
  • Epitaxial thin film growth
  • optimal rate convergence analysis
  • second order backward differentiation formula
  • slope selection

ASJC Scopus subject areas

  • Applied Mathematics

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