Effective time step analysis for the Allen–Cahn equation with a high-order polynomial free energy

Seunggyu Lee; Sungha Yoon; Chaeyoung Lee; Sangkwon Kim; Hyundong Kim; Junxiang Yang; Soobin Kwak; Youngjin Hwang; Junseok Kim

doi:10.1002/nme.7053

Effective time step analysis for the Allen–Cahn equation with a high-order polynomial free energy

Seunggyu Lee, Sungha Yoon, Chaeyoung Lee, Sangkwon Kim, Hyundong Kim, Junxiang Yang, Soobin Kwak, Youngjin Hwang, Junseok Kim

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

7 Citations (Scopus)

Abstract

An effective time step analysis to the linear convex splitting scheme for the Allen–Cahn equation with a high-order polynomial free energy is presented in this article. Although the convex splitting scheme is unconditionally stable, using a large time step causes a time step rescaling effect, leading to delayed dynamics of the governing equation. We verify this problem by comparing it with a reformulated semi-implicit scheme using the effective time step. Theoretical results show that the discrete energy stability and maximum-principle hold, and the numerical results demonstrate that the time step rescaling issue can be resolved using the effective time step. We confirm that slow dynamics due to high-order potential is alleviated by the time step modification through the results of motion by mean curvature.

Original language	English
Pages (from-to)	4726-4743
Number of pages	18
Journal	International Journal for Numerical Methods in Engineering
Volume	123
Issue number	19
DOIs	https://doi.org/10.1002/nme.7053
Publication status	Published - 2022 Oct 15

Bibliographical note

Funding Information:
information Ministry of Education, Grant/Award Number: NRF-2020R1A6A3A13077105; Ministry of Science and ICT, South Korea, Grant/Award Number: 2019R1A6A1A11051177; Ministry of Science, ICT and Future Planning, Grant/Award Number: 2020R1A2C1A01100114The first author (Seunggyu Lee) was supported by a Korea University Grant and the National Research Foundation of Korea (NRF) Grant funded by the Korea government (MSIP) (No. 2020R1A2C1A01100114). The author (Sungha Yoon) was supported by the National Research Foundation of Korea (NRF) Grant funded by the Ministry of Science and ICT of Korea (MSIT) (No. 2019R1A6A1A11051177). The author (Hyundong Kim) was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Republic of Korea (NRF-2020R1A6A3A13077105). The corresponding author (Junseok Kim) was supported by a Grant from the Korea University Grant. The authors are grateful to the reviewers for their contributions to improving this article.

Publisher Copyright:
© 2022 John Wiley & Sons Ltd.

Keywords

Allen–Cahn equation
effective time step
high-order polynomial free energy
linear convex splitting

ASJC Scopus subject areas

Numerical Analysis
General Engineering
Applied Mathematics

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10.1002/nme.7053

Cite this

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title = "Effective time step analysis for the Allen–Cahn equation with a high-order polynomial free energy",

abstract = "An effective time step analysis to the linear convex splitting scheme for the Allen–Cahn equation with a high-order polynomial free energy is presented in this article. Although the convex splitting scheme is unconditionally stable, using a large time step causes a time step rescaling effect, leading to delayed dynamics of the governing equation. We verify this problem by comparing it with a reformulated semi-implicit scheme using the effective time step. Theoretical results show that the discrete energy stability and maximum-principle hold, and the numerical results demonstrate that the time step rescaling issue can be resolved using the effective time step. We confirm that slow dynamics due to high-order potential is alleviated by the time step modification through the results of motion by mean curvature.",

keywords = "Allen–Cahn equation, effective time step, high-order polynomial free energy, linear convex splitting",

author = "Seunggyu Lee and Sungha Yoon and Chaeyoung Lee and Sangkwon Kim and Hyundong Kim and Junxiang Yang and Soobin Kwak and Youngjin Hwang and Junseok Kim",

note = "Funding Information: information Ministry of Education, Grant/Award Number: NRF-2020R1A6A3A13077105; Ministry of Science and ICT, South Korea, Grant/Award Number: 2019R1A6A1A11051177; Ministry of Science, ICT and Future Planning, Grant/Award Number: 2020R1A2C1A01100114The first author (Seunggyu Lee) was supported by a Korea University Grant and the National Research Foundation of Korea (NRF) Grant funded by the Korea government (MSIP) (No. 2020R1A2C1A01100114). The author (Sungha Yoon) was supported by the National Research Foundation of Korea (NRF) Grant funded by the Ministry of Science and ICT of Korea (MSIT) (No. 2019R1A6A1A11051177). The author (Hyundong Kim) was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Republic of Korea (NRF-2020R1A6A3A13077105). The corresponding author (Junseok Kim) was supported by a Grant from the Korea University Grant. The authors are grateful to the reviewers for their contributions to improving this article. Publisher Copyright: {\textcopyright} 2022 John Wiley & Sons Ltd.",

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AU - Kim, Hyundong

AU - Yang, Junxiang

AU - Kwak, Soobin

AU - Hwang, Youngjin

AU - Kim, Junseok

N1 - Funding Information: information Ministry of Education, Grant/Award Number: NRF-2020R1A6A3A13077105; Ministry of Science and ICT, South Korea, Grant/Award Number: 2019R1A6A1A11051177; Ministry of Science, ICT and Future Planning, Grant/Award Number: 2020R1A2C1A01100114The first author (Seunggyu Lee) was supported by a Korea University Grant and the National Research Foundation of Korea (NRF) Grant funded by the Korea government (MSIP) (No. 2020R1A2C1A01100114). The author (Sungha Yoon) was supported by the National Research Foundation of Korea (NRF) Grant funded by the Ministry of Science and ICT of Korea (MSIT) (No. 2019R1A6A1A11051177). The author (Hyundong Kim) was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Republic of Korea (NRF-2020R1A6A3A13077105). The corresponding author (Junseok Kim) was supported by a Grant from the Korea University Grant. The authors are grateful to the reviewers for their contributions to improving this article. Publisher Copyright: © 2022 John Wiley & Sons Ltd.

PY - 2022/10/15

Y1 - 2022/10/15

N2 - An effective time step analysis to the linear convex splitting scheme for the Allen–Cahn equation with a high-order polynomial free energy is presented in this article. Although the convex splitting scheme is unconditionally stable, using a large time step causes a time step rescaling effect, leading to delayed dynamics of the governing equation. We verify this problem by comparing it with a reformulated semi-implicit scheme using the effective time step. Theoretical results show that the discrete energy stability and maximum-principle hold, and the numerical results demonstrate that the time step rescaling issue can be resolved using the effective time step. We confirm that slow dynamics due to high-order potential is alleviated by the time step modification through the results of motion by mean curvature.

AB - An effective time step analysis to the linear convex splitting scheme for the Allen–Cahn equation with a high-order polynomial free energy is presented in this article. Although the convex splitting scheme is unconditionally stable, using a large time step causes a time step rescaling effect, leading to delayed dynamics of the governing equation. We verify this problem by comparing it with a reformulated semi-implicit scheme using the effective time step. Theoretical results show that the discrete energy stability and maximum-principle hold, and the numerical results demonstrate that the time step rescaling issue can be resolved using the effective time step. We confirm that slow dynamics due to high-order potential is alleviated by the time step modification through the results of motion by mean curvature.

KW - Allen–Cahn equation

KW - effective time step

KW - high-order polynomial free energy

KW - linear convex splitting

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VL - 123

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JO - International Journal for Numerical Methods in Engineering

JF - International Journal for Numerical Methods in Engineering

IS - 19

ER -

Effective time step analysis for the Allen–Cahn equation with a high-order polynomial free energy

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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