On the stability and accuracy of partially and fully implicit schemes for phase field modeling

Jinchao Xu, Yukun Li, Shuonan Wu, Arthur Bousquet

Research output: Contribution to journalArticlepeer-review

74 Scopus citations


We study in this paper the accuracy and stability of partially and fully implicit schemes for phase field modeling. Through theoretical and numerical analysis of Allen–Cahn and Cahn–Hilliard models, we investigate the potential problems of using partially implicit schemes, demonstrate the importance of using fully implicit schemes and discuss the limitation of energy stability that are often used to evaluate the quality of a numerical scheme for phase-field modeling. In particular, we make the following observations: 1. a convex splitting scheme (CSS in short) can be equivalent to some fully implicit scheme (FIS in short) with a much different time scaling and thus it may lack numerical accuracy; 2. most implicit schemes are energy-stable if the time-step size is sufficiently small; 3. a traditionally known conditionally energy-stable scheme still possesses an unconditionally energy-stable physical solution; 4. an unconditionally energy-stable scheme is not necessarily better than a conditionally energy-stable scheme when the time step size is not small enough; 5. a first-order FIS for the Allen–Cahn model can be devised so that the maximum principle will be valid on the discrete level and hence the discrete phase variable satisfies |uh(x)|≤1 for all x and, furthermore, the linearized discretized system can be effectively preconditioned by discrete Poisson operators.
Original languageEnglish (US)
Pages (from-to)826-853
Number of pages28
JournalComputer Methods in Applied Mechanics and Engineering
StatePublished - Mar 1 2019
Externally publishedYes

Bibliographical note

Generated from Scopus record by KAUST IRTS on 2023-02-15

ASJC Scopus subject areas

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


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