An efficient method for computing stationary states of phase field crystal models

Kai Jiang School of Mathematics and Computational Science, Xiangtan University, Xiangtan, Hunan, P.R. China, 411105 Wei Si School of Mathematics and Computational Science, Xiangtan University, Xiangtan, Hunan, P.R. China, 411105 Chenglong Bao Yau Mathematical Sciences Center, Tsinghua University, Beijing, P. R. China, 100084

Numerical Analysis and Scientific Computing mathscidoc:2206.25006

2019.9
Computing stationary states is an important topic for phase field crystal (PFC) models. Great efforts have been made for energy dissipation of the numerical schemes when using gradient flows. However, it is always time-consuming due to the requirement of small effective time steps. In this paper, we propose an adaptive accelerated proximal gradient method for finding the stationary states of PFC models. The energy dissipation is guaranteed and the convergence property is established for the discretized energy functional. Moreover, the connections between generalized proximal operator with classical (semi-)implicit and explicit schemes for gradient flow are given. Extensive numerical experiments, including two three dimensional periodic crystals in Landau-Brazovskii (LB) model and a two dimensional quasicrystal in Lifshitz-Petrich (LP) model, demonstrate that our approach has adaptive time steps which lead to significant acceleration over semi-implicit methods for computing complex structures. Furthermore, our result reveals a deep physical mechanism of the simple LB model via which the sigma phase is first discovered.
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@inproceedings{kai2019an,
  title={An efficient method for computing stationary states of phase field crystal models},
  author={Kai Jiang, Wei Si, and Chenglong Bao},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220616155449764495384},
  year={2019},
}
Kai Jiang, Wei Si, and Chenglong Bao. An efficient method for computing stationary states of phase field crystal models. 2019. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220616155449764495384.
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