A practical finite difference scheme for the Navier–Stokes equation on curved surfaces in R3

Junxiang Yang, Yibao Li, Junseok Kim

Research output: Contribution to journalArticlepeer-review

10 Citations (Scopus)


We present a practical finite difference scheme for the incompressible Navier–Stokes equation on curved surfaces in three-dimensional space. In the proposed method, the curved surface is embedded in a narrow band domain and the governing equation is extended to the narrow band domain. We use the standard seven-point stencil for the Laplace operator instead of a discrete Laplacian–Beltrami operator by using the closet point method and pseudo-Neumann boundary condition. The well-known projection method is used to solve the incompressible Navier–Stokes equation in the narrow band domain. To make the velocity field be parallel to the surface, a velocity correction step is used. Various numerical experiments, such as the divergence-free test, the convergence rate, and the energy dissipation, are performed on curved surfaces, which demonstrated that our proposed method is robust and practical.

Original languageEnglish
Article number109403
JournalJournal of Computational Physics
Publication statusPublished - 2020 Jun 15

Bibliographical note

Funding Information:
J. Yang is supported by China Scholarship Council ( 201908260060 ). Y.B. Li is supported by National Natural Science Foundation of China (No. 11601416 , No. 11631012 ). The corresponding author (J.S. Kim) was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education ( NRF-2019R1A2C1003053 ). The authors greatly appreciate the reviewers for their constructive comments and suggestions, which have improved the quality of this paper.

Publisher Copyright:
© 2020 Elsevier Inc.


  • Closest-point method
  • Curved surfaces
  • Incompressible Navier–Stokes equation
  • Narrow band domain
  • Projection method

ASJC Scopus subject areas

  • Numerical Analysis
  • Modelling and Simulation
  • Physics and Astronomy (miscellaneous)
  • General Physics and Astronomy
  • Computer Science Applications
  • Computational Mathematics
  • Applied Mathematics


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