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

  • Xiaobo Jing
  • Jun Li
  • Xueping Zhao
  • Qi WangEmail author


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.


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



This research is partially supported by NSFC Awards #11571032, #91630207 and NSAF-U1530401.


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Authors and Affiliations

  1. 1.Beijing Computational Science Research CenterBeijingP. R. China
  2. 2.School of Mathematical SciencesTianjin Normal UniversityTianjinP. R. China
  3. 3.Department of MathematicsUniversity of South CarolinaColumbiaUSA
  4. 4.Department of MathematicsUniversity of South CarolinaColumbiaUSA
  5. 5.School of Materials Science and EngineeringNankai UniversityTianjinP. R. China

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