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 Kim

doi:10.1007/s10665-019-10023-9

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 Kim

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

4 Citations (Scopus)

Abstract

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.

Original language	English
Pages (from-to)	149-166
Number of pages	18
Journal	Journal of Engineering Mathematics
Volume	119
Issue number	1
DOIs	https://doi.org/10.1007/s10665-019-10023-9
Publication status	Published - 2019 Dec 1

Keywords

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

ASJC Scopus subject areas

Mathematics(all)
Engineering(all)

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10.1007/s10665-019-10023-9

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

abstract = "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.",

keywords = "Closest point method, Conservative scheme, N-component Cahn–Hilliard equation, Narrow band domain",

author = "Junxiang Yang and Yibao Li and Chaeyoung Lee and Darae Jeong and Junseok Kim",

note = "Funding Information: 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. Publisher Copyright: {\textcopyright} 2019, Springer Nature B.V.",

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doi = "10.1007/s10665-019-10023-9",

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AU - Kim, Junseok

N1 - Funding Information: 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. Publisher Copyright: © 2019, Springer Nature B.V.

PY - 2019/12/1

Y1 - 2019/12/1

N2 - 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.

AB - 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.

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KW - Conservative scheme

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KW - Narrow band domain

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

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