An efficient and stable compact fourth-order finite difference scheme for the phase field crystal equation

Yibao Li, Junseok Kim

Research output: Contribution to journalArticlepeer-review

50 Citations (Scopus)


In this paper, we present a high-order accurate compact scheme for the phase field crystal model in two- and three-dimensional spaces. The proposed scheme is derived by combining a fourth-order compact finite difference formula in space and a backward differentiation for the time derivative term, which is second-order accurate in time. Furthermore, a nonlinearly stabilized splitting scheme is used and thus a larger time step can be allowed. Since the equations at the implicit time level are nonlinear, we introduce a Newton-type iterative method and employ a fast and efficient nonlinear multigrid solver to solve the resulting discrete system. In particular, we implement the compact scheme in the adaptive mesh refinement framework. An adaptive time step method for the phase field crystal model is also proposed. Various numerical experiments are presented and confirm the accuracy, stability, and efficiency of our proposed method.

Original languageEnglish
Pages (from-to)194-216
Number of pages23
JournalComputer Methods in Applied Mechanics and Engineering
Publication statusPublished - 2017 Jun 1

Bibliographical note

Funding Information:
Y.B. Li is supported by Natural Science Basic Research Plan in Shaanxi Province of China (2016JQ1024), by National Natural Science Foundation of China (Nos. 11601416, 11631012). The corresponding author (J.S. Kim) was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2016R1D1A1B03933243). The authors are grateful to the reviewers whose valuable suggestions and comments significantly improved the quality of this paper.

Publisher Copyright:
© 2017 Elsevier B.V.


  • Adaptive mesh refinement
  • Adaptive time-stepping
  • Fourth-order compact scheme
  • Phase-field crystal equation

ASJC Scopus subject areas

  • Computational Mechanics
  • Mechanics of Materials
  • Mechanical Engineering
  • General Physics and Astronomy
  • Computer Science Applications


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