A Third Order BDF Energy Stable Linear Scheme for the No-Slope-Selection Thin Film Model (Communications in Computational Physics, 2021)
In this paper we propose and analyze a (temporally) third order accuratebackward differentiation formula (BDF) numerical scheme for the no-slope-selection(NSS) equation of the epitaxial thin film growth model, with Fourier pseudo-spectraldiscretization 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-Dupontregularization term, in the form of $−A∆t^2∆^2_N(u^{n+1}−u^n)$, is added in the numericalscheme. 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, theoptimal 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 theefficiency of the numerical scheme and the third order convergence. The long timesimulation results for $ε = 0.02$ (up to $T = 3×10^5$) have indicated a logarithm law forthe energy decay, as well as the power laws for growth of the surface roughness andthe mound width. In particular, the power index for the surface roughness and themound width growth, created by the third order numerical scheme, is more accuratethan those produced by certain second order energy stable schemes in the existingliterature.
Keywords: Epitaxial thin film growth; no-slope-selection; third order backward differentiation formula; energy stability; optimal rate convergence analysis.