Regularized linear schemes for the molecular beam epitaxy model with slope selection
Regularized linear schemes for the molecular beam epitaxy model with slope selection
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DOI：
10.1016/j.apnum.2018.02.004
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发表时间：
2018-06
影响因子：
2.8
通讯作者：
Lizhen Chen;Jia Zhao;X. Yang;X. Yang
中科院分区：
文献类型：
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作者：
Lizhen Chen;Jia Zhao;X. Yang;X. Yang
本文提出了具有斜率选择的分子束外延(MBE)模型的全离散线性格式，证明了该格式是无条件能量稳定且唯一可解的。具体地说，使用不变能量二次化(IEQ)方法，结合正则化技术，首先使用Crank-Nicolson或Adam-Bashforth策略对MBE模型进行时间离散。证明了半离散格式是能量稳定且唯一可解的。然后利用傅里叶谱方法对空间进行离散化，得到能量稳定且唯一可解的全离散格式。具体地说，全离散格式是线性的，因此在每个时间步只需要解决一个线性代数问题。通过数值试验表明，适当选择正则化参数可以提供更好的稳定性和精度，从而使较大的时间步长是可行的。然后，我们给出了几个数值模拟，以证明我们新提出的方案的准确性和效率。本文所采用的线性化和正则化策略可以方便地用于求解一类由能量变化导出的相场模型。
In this paper, we propose full discrete linear schemes for the molecular beam epitaxy (MBE) model with slope selection, which are shown to be unconditionally energy stable and unique solvable. In details, using the invariant energy quadratization (IEQ) approach, along with a regularized technique, the MBE model is first discretized in time using either Crank–Nicolson or Adam–Bashforth strategies. The semi-discrete schemes are shown to be energy stable and unique solvable. Then we further use Fourier-spectral methods to discretize the space, ending with full discrete schemes that are energy-stable and unique solvable. In particular, the full discrete schemes are linear such that only a linear algebra problem need to be solved at each time step. Through numerical tests, we have shown a proper choice of the regularization parameter provides better stability and accuracy, such that larger time step is feasible. Afterward, we present several numerical simulations to demonstrate the accuracy and efficiency of our newly proposed schemes. The linearizing and regularizing strategy used in this paper could be readily applied to solve a class of phase field models that are derived from energy variation.