A conservative finite difference scheme for the N-component Cahn–Hilliard system on curved surfaces in 3D

  • Junxiang Yang
  • Yibao Li
  • Chaeyoung Lee
  • Darae Jeong
  • Junseok KimEmail author


This paper presents a conservative finite difference scheme for solving the N-component Cahn–Hilliard (CH) system on curved surfaces in three-dimensional (3D) space. Inspired by the closest point method (Macdonald and Ruuth, SIAM J Sci Comput 31(6):4330–4350, 2019), we use the standard seven-point finite difference discretization for the Laplacian operator instead of the Laplacian–Beltrami operator. We only need to independently solve (\(N-1\)) CH equations in a narrow band domain around the surface because the solution for the Nth component can be obtained directly. The N-component CH system is discretized using an unconditionally stable nonlinear splitting numerical scheme, and it is solved by using a Jacobi-type iteration. Several numerical tests are performed to demonstrate the capability of the proposed numerical scheme. The proposed multicomponent model can be simply modified to simulate phase separation in a complex domain on 3D surfaces.


Closest point method Conservative scheme N-component Cahn–Hilliard equation Narrow band domain 



The author (D. Jeong) was supported by the National Research Foundation of Korea (NRF) Grant funded by the Korea government (MSIP) (NRF-2017R1E1A1A03070953). Y.B. Li is supported by National Natural Science Foundation of China (Nos. 11601416, 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-2016R1D1A1B03933243). The authors appreciate the reviewers for their constructive comments, which have improved the quality of this paper.


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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of MathematicsKorea UniversitySeoulRepublic of Korea
  2. 2.School of Mathematics and StatisticsXi’an Jiaotong UniversityXi’anChina
  3. 3.Department of MathematicsKangwon National UniversityGangwon-doRepublic of Korea

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