@Article{CiCP-29-3,
author = {Yonghong, Hao and Qiumei, Huang and Wang, Cheng},
title = {A Third Order BDF Energy Stable Linear Scheme for the No-Slope-Selection Thin Film Model},
journal = {Communications in Computational Physics},
year = {2021},
volume = {29},
number = {3},
pages = {905--929},
abstract = {<p style="text-align: justify;">In this paper we propose and analyze a (temporally) third order accurate
backward differentiation formula (BDF) numerical scheme for the no-slope-selection
(NSS) equation of the epitaxial thin film growth model, with Fourier pseudo-spectral
discretization in space. The surface diffusion term is treated implicitly, while the nonlinear chemical potential is approximated by a third order explicit extrapolation formula for the sake of solvability. In addition, a third order accurate Douglas-Dupont
regularization term, in the form of $−A∆t^2∆^2_N
(u^{n+1}−u^n)$, is added in the numerical
scheme. A careful energy stability estimate, combined with 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 $ℓ^∞(0,T;ℓ^2)∩ℓ^2(0,T;H^2_h)$ norm, with the help of a linearized estimate for the nonlinear error terms. 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.</p>},
issn = {1991-7120},
doi = {https://doi.org/10.4208/cicp.OA-2020-0074},
url = {https://global-sci.com/article/79655/a-third-order-bdf-energy-stable-linear-scheme-for-the-no-slope-selection-thin-film-model}
}