Second Order Linear Energy Stable Schemes for Allen-Cahn Equations with Nonlocal Constraints

Xiaobo Jing, Jun Li, Xueping Zhao, Qi Wang

Research output: Journal PublicationArticlepeer-review

10 Citations (Scopus)


We present a set of linear, second order, unconditionally energy stable schemes for the Allen-Cahn equation with nonlocal constraints that preserves the total volume of each phase in a binary material system. The energy quadratization strategy is employed to derive the energy stable semi-discrete numerical algorithms in time. Solvability conditions are then established for the linear systems resulting from the semi-discrete, linear schemes. The fully discrete schemes are obtained afterwards by applying second order finite difference methods on cell-centered grids in space. The performance of the schemes are assessed against two benchmark numerical examples, in which dynamics obtained using the volume-preserving Allen-Cahn equations with nonlocal constraints is compared with those obtained using the classical Allen-Cahn as well as the Cahn-Hilliard model, respectively, demonstrating slower dynamics when volume constraints are imposed as well as their usefulness as alternatives to the Cahn–Hilliard equation in describing phase evolutionary dynamics for immiscible material systems while preserving the phase volumes. Some performance enhancing, practical implementation methods for the linear energy stable schemes are discussed in the end.

Original languageEnglish
Pages (from-to)500-537
Number of pages38
JournalJournal of Scientific Computing
Issue number1
Publication statusPublished - 15 Jul 2019
Externally publishedYes


  • Energy quadratization
  • Energy stable schemes
  • Nonlocal constraints
  • Phase field model
  • Volume preserving

ASJC Scopus subject areas

  • Software
  • Theoretical Computer Science
  • Numerical Analysis
  • General Engineering
  • Computational Theory and Mathematics
  • Computational Mathematics
  • Applied Mathematics


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