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$$∆^2_N$($u^{n+1}$−$u^n$), is added in the numerical scheme to justify
the energy stability at a theoretical level. Due to an alternate expression of the nonlinear chemical potential terms, such a numerical scheme requires a minimum value of
the artificial regularization parameter as A=$\frac{289}{1024}$, 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,$T$;$ℓ^2$)∩$ℓ^2$(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 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 the
power laws for growth of the surface roughness and the mound width.