Efficient Numerical Methods for Computing the Stationary States of Phase Field Crystal Models

Kai Jiang School of Mathematics and Computational Science, Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University, Xiangtan, Hunan, China, 411105 Wei Si School of Mathematics and Computational Science, Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University, Xiangtan, Hunan, China, 411105 Chang Chen School of Mathematics and Computational Science, Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University, Xiangtan, Hunan, China, 411105 Chenglong Bao Yau Mathematical Sciences Center, Tsinghua University, Beijing, China, 100084

Numerical Analysis and Scientific Computing mathscidoc:2206.25002

SIAM J. SCI. COMPUT., 42, (6), 2020.11
Finding the stationary states of a free energy functional is an important problem in phase field crystal (PFC) models. Many efforts have been devoted to designing numerical schemes with energy dissipation and mass conservation properties. However, most existing approaches are time-consuming due to the requirement of small effective step sizes. In this paper, we discretize the energy functional and propose efficient numerical algorithms for solving the constrained nonconvex minimization problem. A class of gradient-based approaches, which are the so-called adaptive accelerated Bregman proximal gradient (AA-BPG) methods, is proposed, and the convergence property is established without the global Lipschitz constant requirements. A practical Newton method is also designed to further accelerate the local convergence with convergence guarantee. One key feature of our algorithms is that the energy dissipation and mass conservation properties hold during the iteration process. Moreover, we develop a hybrid acceleration framework to accelerate the AA-BPG methods and most of the existing approaches through coupling with the practical Newton method. Extensive numerical experiments, including two three-dimensional periodic crystals in the Landau--Brazovskii (LB) model and a two-dimensional quasicrystal in the Lifshitz--Petrich (LP) model, demonstrate that our approaches have adaptive step sizes which lead to a significant acceleration over many existing methods when computing complex structures.
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@inproceedings{kai2020efficient,
  title={Efficient Numerical Methods for Computing the Stationary States of Phase Field Crystal Models},
  author={Kai Jiang, Wei Si, Chang Chen, and Chenglong Bao},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220613221531417894355},
  booktitle={SIAM J. SCI. COMPUT.},
  volume={42},
  number={6},
  year={2020},
}
Kai Jiang, Wei Si, Chang Chen, and Chenglong Bao. Efficient Numerical Methods for Computing the Stationary States of Phase Field Crystal Models. 2020. Vol. 42. In SIAM J. SCI. COMPUT.. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220613221531417894355.
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