论文标题
无斜率选择薄膜模型的三阶BDF能量稳定线性方案
A third order BDF energy stable linear scheme for the no-slope-selection thin film model
论文作者
论文摘要
在本文中，我们提出和分析了一个（暂时）的三阶准确向后分化公式（BDF）数值方案，用于在空间中具有傅立叶薄膜生长模型的No-Selection（NSS）方程（NSS）方程。表面扩散项被隐式处理，而非线性化学势则通过三阶显式外推公式近似，以实现溶解度。此外，在数值方案中添加了$-AδT^2δ_n^2（u^{n+1} -U^n）$的三阶准确的Douglas-Dupont正则化项。仔细的能量稳定性估计，结合傅立叶特征值分析，在修改版本中导致能量稳定性，并提供了系数$ a $的理论理由。由于这种能量稳定性分析，获得了数值能量的统一。此外，最佳速率收敛分析和误差估计是在$ \ ell^\ infty（0，t; \ ell^2）\ cap \ ell^2（0，t; h_h^2）$ norm的详细信息中得出的，借助非线性误差项的线性化估计值。 ％此收敛估计是梯度流的三阶准确方案的第一个结果。提出了一些数值模拟结果，以证明数值方案的效率和三阶收敛。 $ \ varepsilon的长期模拟结果= 0.02 $（最多$ t = 3 \ times 10^5 $）表示能量衰减的对数法律以及表面粗糙度和土墩宽度的生长的功率定律。特别是，比三阶数值方案创建的表面粗糙度和土墩宽度生长的功率指数比现有文献中某些二阶能量稳定方案产生的功率指数更准确。
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 $-A Δt^2 Δ_N^2 ( u^{n+1} - u^n)$, 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 $\ell^\infty (0,T; \ell^2) \cap \ell^2 (0,T; H_h^2)$ norm, with the help of a linearized estimate for the nonlinear error terms. %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 $\varepsilon=0.02$ (up to $T=3 \times 10^5$) 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.
