Volume 1, Issue 3
Artificial Regularization Parameter Analysis for the No-Slope-Selection Epitaxial Thin Film Model

Xiangjun Meng, Zhonghua Qiao, Cheng Wang & Zhengru Zhang

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

Published online: 2020-09

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  • Abstract

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.

  • Keywords

Epitaxial thin film growth, slope selection, second order backward differentiation formula, energy stability, Douglas-Dupont regularization, optimal rate convergence analysis.

  • AMS Subject Headings

35K30, 35K55, 65L06, 65M12, 65M70, 65T40

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CSIAM-AM-1-441, author = {Xiangjun Meng , and Zhonghua Qiao , and Cheng Wang , and Zhengru Zhang , }, title = {Artificial Regularization Parameter Analysis for the No-Slope-Selection Epitaxial Thin Film Model}, journal = {CSIAM Transactions on Applied Mathematics}, year = {2020}, volume = {1}, number = {3}, pages = {441--462}, abstract = {

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} }
TY - JOUR T1 - Artificial Regularization Parameter Analysis for the No-Slope-Selection Epitaxial Thin Film Model AU - Xiangjun Meng , AU - Zhonghua Qiao , AU - Cheng Wang , AU - Zhengru Zhang , JO - CSIAM Transactions on Applied Mathematics VL - 3 SP - 441 EP - 462 PY - 2020 DA - 2020/09 SN - 1 DO - http://doi.org/10.4208/csiam-am.2020-0015 UR - https://global-sci.org/intro/article_detail/csiam-am/18302.html KW - Epitaxial thin film growth, slope selection, second order backward differentiation formula, energy stability, Douglas-Dupont regularization, optimal rate convergence analysis. AB -

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.

Xiangjun Meng, Zhonghua Qiao, Cheng Wang & Zhengru Zhang. (2020). Artificial Regularization Parameter Analysis for the No-Slope-Selection Epitaxial Thin Film Model. CSIAM Transactions on Applied Mathematics. 1 (3). 441-462. doi:10.4208/csiam-am.2020-0015
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