Long-time simulation of the phase-field crystal equation using high-order energy-stable CSRK methods

Jaemin Shin, Hyun Geun Lee, June Yub Lee

Research output: Contribution to journalArticlepeer-review

10 Scopus citations

Abstract

The phase-field crystal (PFC) equation is derived by the gradient flow for the Swift–Hohenberg free energy functional; thus, the numerical method requires the energy of the functional to decrease. Convex Splitting Runge–Kutta (CSRK) methods can be suitably applied to achieve high-order temporal accuracy as well as unconditional energy stability and unique solvability. For the PFC equation, we prove the unconditional energy stability and unique solvability of the CSRK methods and provide one family of parameters of the second-order CSRK methods and possible examples of third-order CSRK methods. We present numerical experiments to demonstrate the accuracy and energy stability of the methods. Specifically, based on the high-order accuracy and energy stability of the CSRK method, we propose an indicator function capable of characterizing the pattern formation of the phase-field crystal model for long-time simulation.

Original languageEnglish
Article number112981
JournalComputer Methods in Applied Mechanics and Engineering
Volume364
DOIs
StatePublished - 1 Jun 2020

Keywords

  • Convex Splitting Runge–Kutta method
  • High-order temporal accuracy
  • Long-time simulation
  • Phase-field crystal equation
  • Unconditional energy stability

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