快速显式算子分裂方法在分子束外延模型中的收敛性
Convergence of a Fast Explicit Operator Splitting Method for the Molecular Beam Epitaxy Model
AI总结：
本文提出一种快速显式算子分裂方法，用于分子束外延模型，通过伪谱方法和三十三点差分方案解决线性和非线性子问题，并分析了算法的收敛性，证明了全局误差阶为O(τ²+h⁴)。
摘要：
本文介绍了用于分子束外延模型的快速显式算子分裂（FEOS）方法。原问题被分为线性和非线性子问题。对于线性部分，采用伪谱方法；对于非线性部分，构造了一个点数更少的紧中心差分方案。此外，分析了该算法的收敛率。在离散L²范数下，理论证明了全局误差阶为O(τ²+h⁴)，并通过数值实验验证。一些数值实验显示，该算法在一维情况下对小四阶项系数具有鲁棒性。此外，在大域域中模拟了粗化动力学，并在二维情况下观察到1/3次方律。
英文摘要：
A fast explicit operator splitting (FEOS) method for the molecular beam epitaxy model has been presented in [Cheng, et al., Fast and stable explicit operator splitting methods for phase-field models, J. Comput. Phys., submitted]. The original problem is split into linear and nonlinear subproblems. For the linear part, the pseudo-spectral method is adopted; for the nonlinear part, a 33-point difference scheme is constructed. Here, we give a compact center-difference scheme involving fewer points for the nonlinear subproblem. Besides, we analyze the convergence rate of the algorithm. The global error order $\mathcal{O}(τ^2+h^4)$ in discrete $L^2$-norm is proved theoretically and verified numerically. Some numerical experiments show the robustness of the algorithm for small coefficients of the fourth-order term for the one-dimensional case. Besides, coarsening dynamics are simulated in large domains and the $1/3$ power laws are observed for the two-dimensional case.
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