Artificial Regularization Parameter Analysis for the No-Slope-Selection Epitaxial Thin Film Model (CSIAM Transactions on Applied Mathematics, 2020)
In this paper we study the effect of the artificial regularization term for thesecond 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, withFourier 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 regularizationterm, in the form of −$A$∆$t$$∆^2_N$($u^{n+1}$−$u^n$), is added in the numerical scheme to justifythe energy stability at a theoretical level. Due to an alternate expression of the nonlinear chemical potential terms, such a numerical scheme requires a minimum value ofthe artificial regularization parameter as A=$\frac{289}{1024}$, much smaller than the other reportedartificial parameter values in the existing literature. Such an optimization of the artificial parameter value is expected to reduce the numerical diffusion, and henceforthimprove the long time numerical accuracy. Moreover, the optimal rate convergenceanalysis 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 numericalsimulation 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×$10^5$) have indicated a logarithm law for the energy decay, as well as thepower laws for growth of the surface roughness and the mound width.
Keywords: Epitaxial thin film growth; slope selection; second order backward differentiation formula; energy stability; Douglas-Dupont regularization; optimal rate convergence analysis.