Energy stability and error estimates of exponential time differencing schemes for the epitaxial growth model without slope selection
Energy stability and error estimates of exponential time differencing schemes for the epitaxial growth model without slope selection
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DOI：
10.1090/mcom/3262
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发表时间：
2018
期刊：
影响因子：
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通讯作者：
L. Ju;Xiao Li;Zhonghua Qiao;Hui Zhang
中科院分区：
文献类型：
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作者：
L. Ju;Xiao Li;Zhonghua Qiao;Hui Zhang
本文提出了一类求解无斜率选择的外延生长模型的指数时间差分格式。首先对模型的能量泛函进行线性凸分裂，然后分别采用Fourier配点法和ETD法对相应的梯度流方程进行空间离散和时间积分。在完全离散的意义下，严格地建立了一阶和二阶ETD格式的能量稳定性和误差估计。我们还数值证明了所提出的计划的准确性和模拟的粗化动力学与小扩散系数。结果表明，能量衰减的对数律和幂律增长的表面粗糙度和土堆宽度，这与现有的理论文献中一致。
In this paper, we propose a class of exponential time differencing (ETD) schemes for solving the epitaxial growth model without slope selection. A linear convex splitting is first applied to the energy functional of the model, and then Fourier collocation and ETD-based multistep approximations are used respectively for spatial discretization and time integration of the corresponding gradient flow equation. Energy stabilities and error estimates of the first and second order ETD schemes are rigorously established in the fully discrete sense. We also numerically demonstrate the accuracy of the proposed schemes and simulate the coarsening dynamics with small diffusion coefficients. The results show the logarithm law for the energy decay and the power laws for growth of the surface roughness and the mound width, which are consistent with the existing theories in the literature.