Volume 29, Issue 3
A Third Order BDF Energy Stable Linear Scheme for the No-Slope-Selection Thin Film Model

Yonghong Hao, Qiumei HuangCheng Wang

Commun. Comput. Phys., 29 (2021), pp. 905-929.

Published online: 2021-01

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  • Abstract

In this paper we propose and analyze a (temporally) third order accurate backward differentiation formula (BDF) numerical scheme for the no-slope-selection (NSS) equation of the epitaxial thin film growth model, with Fourier pseudo-spectral discretization 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-Dupont regularization term, in the form of $−A∆t^2∆^2_N (u^{n+1}−u^n)$, is added in the numerical scheme. 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, 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 of the numerical scheme and the third order convergence. 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. In particular, the power index for the surface roughness and the mound width growth, created by the third order numerical scheme, is more accurate than those produced by certain second order energy stable schemes in the existing literature.

  • Keywords

Epitaxial thin film growth, no-slope-selection, third order backward differentiation formula, energy stability, optimal rate convergence analysis.

  • AMS Subject Headings

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

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-29-905, author = {Hao , Yonghong and Huang , Qiumei and Wang , Cheng}, title = {A Third Order BDF Energy Stable Linear Scheme for the No-Slope-Selection Thin Film Model}, journal = {Communications in Computational Physics}, year = {2021}, volume = {29}, number = {3}, pages = {905--929}, abstract = {

In this paper we propose and analyze a (temporally) third order accurate backward differentiation formula (BDF) numerical scheme for the no-slope-selection (NSS) equation of the epitaxial thin film growth model, with Fourier pseudo-spectral discretization 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-Dupont regularization term, in the form of $−A∆t^2∆^2_N (u^{n+1}−u^n)$, is added in the numerical scheme. 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, 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 of the numerical scheme and the third order convergence. 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. In particular, the power index for the surface roughness and the mound width growth, created by the third order numerical scheme, is more accurate than those produced by certain second order energy stable schemes in the existing literature.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2020-0074}, url = {http://global-sci.org/intro/article_detail/cicp/18570.html} }
TY - JOUR T1 - A Third Order BDF Energy Stable Linear Scheme for the No-Slope-Selection Thin Film Model AU - Hao , Yonghong AU - Huang , Qiumei AU - Wang , Cheng JO - Communications in Computational Physics VL - 3 SP - 905 EP - 929 PY - 2021 DA - 2021/01 SN - 29 DO - http://doi.org/10.4208/cicp.OA-2020-0074 UR - https://global-sci.org/intro/article_detail/cicp/18570.html KW - Epitaxial thin film growth, no-slope-selection, third order backward differentiation formula, energy stability, optimal rate convergence analysis. AB -

In this paper we propose and analyze a (temporally) third order accurate backward differentiation formula (BDF) numerical scheme for the no-slope-selection (NSS) equation of the epitaxial thin film growth model, with Fourier pseudo-spectral discretization 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-Dupont regularization term, in the form of $−A∆t^2∆^2_N (u^{n+1}−u^n)$, is added in the numerical scheme. 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, 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 of the numerical scheme and the third order convergence. 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. In particular, the power index for the surface roughness and the mound width growth, created by the third order numerical scheme, is more accurate than those produced by certain second order energy stable schemes in the existing literature.

Yonghong Hao, Qiumei Huang & Cheng Wang. (2021). A Third Order BDF Energy Stable Linear Scheme for the No-Slope-Selection Thin Film Model. Communications in Computational Physics. 29 (3). 905-929. doi:10.4208/cicp.OA-2020-0074
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