An energy stable, hexagonal finite difference scheme for the 2D phase field crystal amplitude equations

Zhen Guan, Vili Heinonen, John Lowengrub, Cheng Wang, Steven M. Wise*

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

15 Citations (Scopus)
34 Downloads (Pure)


In this paper we construct an energy stable finite difference scheme for the amplitude expansion equations for the two-dimensional phase field crystal (PFC) model. The equations are formulated in a periodic hexagonal domain with respect to the reciprocal lattice vectors to achieve a provably unconditionally energy stable and solvable scheme. To our knowledge, this is the first such energy stable scheme for the PFC amplitude equations. The convexity of each part in the amplitude equations is analyzed, in both the semi-discrete and fully-discrete cases. Energy stability is based on a careful convexity analysis for the energy (in both the spatially continuous and discrete cases). As a result, unique solvability and unconditional energy stability are available for the resulting scheme. Moreover, we show that the scheme is point-wise stable for any time and space step sizes. An efficient multigrid solver is devised to solve the scheme, and a few numerical experiments are presented, including grain rotation and shrinkage and grain growth studies, as examples of the strength and robustness of the proposed scheme and solver.

Original languageEnglish
Pages (from-to)1026-1054
Number of pages29
JournalJournal of Computational Physics
Publication statusPublished - 15 Sept 2016
MoE publication typeA1 Journal article-refereed


  • Amplitude equations
  • Energy stable scheme
  • Hexagonal finite differences
  • Multigrid
  • Phase field crystal


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