无斜率选择薄膜模型的三阶BDF能量稳定线性格式
A third order BDF energy stable linear scheme for the no-slope-selection thin film model
摘要：
本文提出并分析了外延薄膜生长模型的无斜率选择（NSS）方程的一种（时间）三阶精度的后向微分公式（BDF）数值格式，并在空间上进行了傅立叶伪谱离散。表面扩散项被隐式处理，而非线性化学势近似的三阶显式外推公式，为了可解性。此外，在数值格式中加入了一个三阶精度的Douglas-Dupont正则化项，其形式为$-A Δt^2 Δ_N^2 ( u^{n+1} - u^n)$。一个仔细的能量稳定性估计，结合傅立叶特征值分析，在一个修改后的版本中的能量稳定性的结果，和系数$A$的理论理由变得可用。作为能量稳定性分析的结果，得到了数值能量在时间上的统一界。在$\ell^\infty (0,T; \ell^2) \cap \ell^2 (0,T; H_h^2)$范数下，借助于非线性误差项的线性化估计，详细地给出了最优速率收敛分析和误差估计。这一收敛估计是梯度流三阶精度格式的第一个这样的结果。通过数值模拟，验证了数值格式的有效性和三阶收敛性。对$\varepsilon=0.02$（直至$T=3 \times 10^5$）的长时间模拟结果表明，能量衰减呈对数规律，表面粗糙度和丘宽增长呈幂律。特别是，功率指数的表面粗糙度和土堆宽度增长，创建的三阶数值方案，是更准确的比某些二阶能量稳定的计划在现有文献中产生的。
英文摘要：
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.
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