CSIAM Trans. Appl. Math., 1 (2020), pp. 441-462.

Published online: 2020-09

- BibTex
- RIS
- TXT

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.

}, issn = {2708-0579}, doi = {https://doi.org/10.4208/csiam-am.2020-0015}, url = {http://global-sci.org/intro/article_detail/csiam-am/18302.html} }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.

*CSIAM Transactions on Applied Mathematics*.

*1*(3). 441-462. doi:10.4208/csiam-am.2020-0015