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Advances in Computational Mathematics

, Volume 45, Issue 3, pp 1551–1580 | Cite as

Efficient numerical schemes with unconditional energy stabilities for the modified phase field crystal equation

  • Qi Li
  • Liquan MeiEmail author
  • Xiaofeng Yang
  • Yibao Li
Article

Abstract

We consider numerical approximations for the modified phase field crystal equation in this paper. The model is a nonlinear damped wave equation that includes both diffusive dynamics and elastic interactions. To develop easy-to-implement time-stepping schemes with unconditional energy stabilities, we adopt the “Invariant Energy Quadratization” approach. By using the first-order backward Euler, the second-order Crank–Nicolson, and the second-order BDF2 formulas, we obtain three linear and symmetric positive definite schemes. We rigorously prove their unconditional energy stabilities and implement a number of 2D and 3D numerical experiments to demonstrate the accuracy, stability, and efficiency.

Keywords

Modified phase field crystal equation Unconditionally energy stable Pseudo energy Invariant energy quadratization 

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Notes

Acknowledgments

The first author would like to thank Zhen Xu at Beijing Normal University and Yali Gao at Northwestern Polytechnical University for the valuable discussions. The authors also would like to thank the reviewers for their helpful comments and suggestions.

Funding information

The work of Q. Li is supported by the China Scholarship Council (CSC No. 201806280137). The work of L. Mei is partially supported by the NSFC under grant no. 11371289. The work of X. Yang is partially supported by the NSF Grant DMS-1720212 and USC ASPIRE I Track-III/IV Fund. The work of Y. Li is partially supported by the NSFC under grant no. 11601416.

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsXi’an Jiaotong UniversityXi’anPeople’s Republic of China
  2. 2.Department of MathematicsUniversity of South CarolinaColumbiaUSA

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