A Third Order Exponential Time Differencing Numerical Scheme for No-Slope-Selection Epitaxial ...

发布时间:2022-06-29 | 发布人:知识服务与技术支持部

标题: A Third Order Exponential Time Differencing Numerical Scheme for No-Slope-Selection Epitaxial Thin Film Model with Energy Stability

作者: Cheng, KelongQiao, ZhonghuaWang, Cheng

来源出版物: JOURNAL OF SCIENTIFIC COMPUTING

出版时间: OCT 2019

作者关键词: Epitaxial thin film growthSlope selectionExponential time differencingEnergy stabilityOptimal rate convergence analysisAliasing error

研究方向: Mathematics

第一地址: 西南科技大学

入藏号:WOS:000485319000009

中国科学院文献情报中心期刊分区: 数学2

摘要:

In this paper we propose and analyze a (temporally) third order accurate exponential time differencing (ETD) numerical scheme for the no-slope-selection (NSS) equation of the epitaxial thin film growth model, with Fourier pseudo-spectral discretization in space. A linear splitting is applied to the physical model, and an ETD-based multistep approximation is used for time integration of the corresponding equation. In addition, a third order accurate Douglas-Dupont regularization term, in the form of -A Delta t(2)phi(0)(L-N)Delta(2)(N) (u(n+1) - u(n)), is added in the numerical scheme. A careful 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(infinity)(0, T; H-h(1)) boolean AND l(2)(0, T; H-h(3)) norm, with the help of a careful eigenvalue bound estimate, combined with the nonlinear analysis for the NSS model. This convergence estimate is the first such result for a third order accurate scheme for a gradient flow. 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 epsilon = 0.02 (up to T = 3 x 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.