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

Abstract

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∆t 2 φ 0 (L N )∆ 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 ∞ (0, T ; H 1 h )∩ℓ 2 (0, T ; H 3 h ) 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 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.

Artificial Regularization Parameter Analysis for the No-Slope-Selection Epitaxial Thin Film Model

“…For instance, the saturation time scale for the numerical results with A = 0.1 is around 3×10 5 , while the one with A = 289 1024 is around 5×10 5 . By the numerical simulation of a third order accurate ETD-based numerical scheme reported in a recent work [9], we believe that the saturation time scale of 3×10 5 would be more accurate. And also, the detailed numerical data show that long time asymptotic growth rate of the standard deviation given by the regularization pa- 1024 , as recorded in the two figures.…”

A second order accuracy in time, Fourier pseudo-spectral numerical scheme for "Good" Boussinesq equation

“…The proof can be found in [22]; also see the related analysis works in [8,9,11,12,13,14,21,25,35,36], etc.…”

Tikhonov Regularization Terms for Accelerating Inertial Mann-Like Algorithm with Applications

“…To various gradient flows, the artificial regularization term has played an essential role in the energy stability analysis such as epitaxial thin film model either with or without slope selection, a second order energy stable BDF method for the epitaxial thin film equation with slope selection, and a third order exponential time differencing numerical scheme for no-slope-selection epitaxial thin film model. For more details about this topic, see [22][23][24][25][26][27][28].…”