Explicit Hybrid Numerical Method for the Allen-Cahn Type Equations on Curved Surfaces

Yongho Choi, Yibao Li, Chaeyoung Lee, Hyundong Kim, Junseok Kim

Research output: Contribution to journalArticlepeer-review

3 Citations (Scopus)


We present a simple and fast explicit hybrid numerical scheme for the motion by mean curvature on curved surfaces in three-dimensional (3D) space. We numerically solve the Allen-Cahn (AC) and conservative Allen-Cahn (CAC) equations on a triangular surface mesh. We use the operator splitting method and an explicit hybrid numerical method. For the AC equation, we solve the diffusion term using a discrete Laplace-Beltrami operator on the triangular surface mesh and solve the reaction term using the closed-form solution, which is obtained using the separation of variables. Next, for the CAC equation, we additionally solve the time-space dependent Lagrange multiplier using an explicit scheme. Our numerical scheme is computationally fast and efficient because we use an explicit hybrid numerical scheme. We perform various numerical experiments to demonstrate the robustness and efficiency of the proposed scheme.

Original languageEnglish
Pages (from-to)797-810
Number of pages14
JournalNumerical Mathematics
Issue number3
Publication statusPublished - 2021 Jun

Bibliographical note

Publisher Copyright:
©2021 Global-Science Press


  • Allen-Cahn equation
  • Conservative Allen-Cahn equation
  • Hybrid numerical method
  • Laplace-Beltrami operator
  • PDE on surface
  • Triangular surface mesh

ASJC Scopus subject areas

  • Modelling and Simulation
  • Control and Optimization
  • Computational Mathematics
  • Applied Mathematics


Dive into the research topics of 'Explicit Hybrid Numerical Method for the Allen-Cahn Type Equations on Curved Surfaces'. Together they form a unique fingerprint.

Cite this