Linear, Second-Order Accurate, and Energy Stable Scheme for a Ternary Cahn–Hilliard Model by Using Lagrange Multiplier Approach

Junxiang Yang; Junseok Kim

doi:10.1007/s10440-021-00405-6

Linear, Second-Order Accurate, and Energy Stable Scheme for a Ternary Cahn–Hilliard Model by Using Lagrange Multiplier Approach

Junxiang Yang, Junseok Kim

Department of Mathematics

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

10 Citations (Scopus)

Abstract

We develop a second-order accurate, energy stable, and linear numerical method for a ternary Cahn–Hilliard (CH) model. The proposed scheme is an extension of typical Lagrange multiplier approach for binary CH system. The second-order backward difference formula (BDF2) is applied to construct time discretization. We theoretically prove the mass conservation, unique solvability, and energy stability of the proposed scheme. We efficiently solve the resulting discrete linear system by using a multigrid algorithm. The numerical solutions demonstrate that the proposed scheme is practically stable and second-order accurate in time and space. Moreover, we can use the proposed scheme as an effective solver to calculate the ternary CH equations in ternary phase-field fluid systems.

Original language	English
Article number	10
Journal	Acta Applicandae Mathematicae
Volume	172
Issue number	1
DOIs	https://doi.org/10.1007/s10440-021-00405-6
Publication status	Published - 2021 Apr

Bibliographical note

Funding Information:
J. Yang is supported by China Scholarship Council (201908260060). 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).

Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer Nature B.V.

Keywords

Energy stable scheme
Multigrid method
Second-order accuracy
Ternary Cahn–Hilliard model

ASJC Scopus subject areas

Applied Mathematics

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10.1007/s10440-021-00405-6

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abstract = "We develop a second-order accurate, energy stable, and linear numerical method for a ternary Cahn–Hilliard (CH) model. The proposed scheme is an extension of typical Lagrange multiplier approach for binary CH system. The second-order backward difference formula (BDF2) is applied to construct time discretization. We theoretically prove the mass conservation, unique solvability, and energy stability of the proposed scheme. We efficiently solve the resulting discrete linear system by using a multigrid algorithm. The numerical solutions demonstrate that the proposed scheme is practically stable and second-order accurate in time and space. Moreover, we can use the proposed scheme as an effective solver to calculate the ternary CH equations in ternary phase-field fluid systems.",

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AU - Yang, Junxiang

AU - Kim, Junseok

N1 - Funding Information: J. Yang is supported by China Scholarship Council (201908260060). 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). Publisher Copyright: © 2021, The Author(s), under exclusive licence to Springer Nature B.V.

PY - 2021/4

Y1 - 2021/4

N2 - We develop a second-order accurate, energy stable, and linear numerical method for a ternary Cahn–Hilliard (CH) model. The proposed scheme is an extension of typical Lagrange multiplier approach for binary CH system. The second-order backward difference formula (BDF2) is applied to construct time discretization. We theoretically prove the mass conservation, unique solvability, and energy stability of the proposed scheme. We efficiently solve the resulting discrete linear system by using a multigrid algorithm. The numerical solutions demonstrate that the proposed scheme is practically stable and second-order accurate in time and space. Moreover, we can use the proposed scheme as an effective solver to calculate the ternary CH equations in ternary phase-field fluid systems.

AB - We develop a second-order accurate, energy stable, and linear numerical method for a ternary Cahn–Hilliard (CH) model. The proposed scheme is an extension of typical Lagrange multiplier approach for binary CH system. The second-order backward difference formula (BDF2) is applied to construct time discretization. We theoretically prove the mass conservation, unique solvability, and energy stability of the proposed scheme. We efficiently solve the resulting discrete linear system by using a multigrid algorithm. The numerical solutions demonstrate that the proposed scheme is practically stable and second-order accurate in time and space. Moreover, we can use the proposed scheme as an effective solver to calculate the ternary CH equations in ternary phase-field fluid systems.

KW - Energy stable scheme

KW - Multigrid method

KW - Second-order accuracy

KW - Ternary Cahn–Hilliard model

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VL - 172

JO - Acta Applicandae Mathematicae

JF - Acta Applicandae Mathematicae

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ER -

Linear, Second-Order Accurate, and Energy Stable Scheme for a Ternary Cahn–Hilliard Model by Using Lagrange Multiplier Approach

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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