Journal of Scientific Computing

, Volume 79, Issue 1, pp 593–623 | Cite as

Error Estimates of Energy Stable Numerical Schemes for Allen–Cahn Equations with Nonlocal Constraints

  • Shouwen Sun
  • Xiaobo Jing
  • Qi WangEmail author


We present error estimates for four unconditionally energy stable numerical schemes developed for solving Allen–Cahn equations with nonlocal constraints. The schemes are linear and second order in time and space, designed based on the energy quadratization (EQ) or the scalar auxiliary variable (SAV) method, respectively. In addition to the rigorous error estimates for each scheme, we also show that the linear systems resulting from the energy stable numerical schemes are all uniquely solvable. Then, we present some numerical experiments to show the accuracy of the schemes, their volume-preserving as well as energy dissipation properties in a drop merging simulation.


Energy stable schemes Energy quadratization methods Scalar auxiliary variable methods Error estimates Finite difference methods 



Qi Wang’s research is partially supported by NSF awards DMS-1517347, DMS-1815921 and OIA-1655740, and NSFC awards #11571032, #91630207 and NSAF-U1530401.


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Authors and Affiliations

  1. 1.Beijing Computational Science Research CenterBeijingPeople’s Republic of China
  2. 2.Department of MathematicsUniversity of South CarolinaColumbiaUSA
  3. 3.School of Materials Science and EngineeringNankai UniversityTianjinChina

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