On the evolutionary dynamics of the cahn-hilliard equation with cut-off mass source

Chaeyoung Lee; Hyundong Kim; Sungha Yoon; Jintae Park; Sangkwon Kim; Junxiang Yang; Junseok Kim

doi:10.4208/NMTMA.OA-2020-0051

On the evolutionary dynamics of the cahn-hilliard equation with cut-off mass source

Chaeyoung Lee, Hyundong Kim, Sungha Yoon, Jintae Park, Sangkwon Kim, Junxiang Yang, Junseok Kim

Department of Mathematics

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

2 Citations (Scopus)

Abstract

We investigate the effect of cut-off logistic source on evolutionary dynamics of a generalized Cahn-Hilliard (CH) equation in this paper. It is a well-known fact that the maximum principle does not hold for the CH equation. Therefore, a generalized CH equation with logistic source may cause the negative concentration blow-up problem in finite time. To overcome this drawback, we propose the cut-off logistic source such that only the positive value greater than a given critical concentration can grow. We consider the temporal profiles of numerical results in the one-, two-, and three-dimensional spaces to examine the effect of extra mass source. Numerical solutions are obtained using a finite difference multigrid solver. Moreover, we perform numerical tests for tumor growth simulation, which is a typical application of generalized CH equations in biology. We apply the proposed cut-off logistic source term and have good results.

Original language	English
Pages (from-to)	242-260
Number of pages	19
Journal	Numerical Mathematics
Volume	14
Issue number	1
DOIs	https://doi.org/10.4208/NMTMA.OA-2020-0051
Publication status	Published - 2020 Oct

Bibliographical note

Funding Information:
The first author (C. Lee) was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2019R1A6A3A13094308). 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 thank the editor and the reviewers for their constructive and helpful comments on the revision of this article.

Publisher Copyright:
© 2021 Global-Science Press.

Keywords

Cahn-Hilliard equation
Finite difference method
Logistic source
Tumor growth application

ASJC Scopus subject areas

Modelling and Simulation
Control and Optimization
Computational Mathematics
Applied Mathematics

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title = "On the evolutionary dynamics of the cahn-hilliard equation with cut-off mass source",

abstract = "We investigate the effect of cut-off logistic source on evolutionary dynamics of a generalized Cahn-Hilliard (CH) equation in this paper. It is a well-known fact that the maximum principle does not hold for the CH equation. Therefore, a generalized CH equation with logistic source may cause the negative concentration blow-up problem in finite time. To overcome this drawback, we propose the cut-off logistic source such that only the positive value greater than a given critical concentration can grow. We consider the temporal profiles of numerical results in the one-, two-, and three-dimensional spaces to examine the effect of extra mass source. Numerical solutions are obtained using a finite difference multigrid solver. Moreover, we perform numerical tests for tumor growth simulation, which is a typical application of generalized CH equations in biology. We apply the proposed cut-off logistic source term and have good results. ",

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note = "Funding Information: The first author (C. Lee) was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2019R1A6A3A13094308). 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 thank the editor and the reviewers for their constructive and helpful comments on the revision of this article. Publisher Copyright: {\textcopyright} 2021 Global-Science Press.",

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AU - Kim, Sangkwon

AU - Yang, Junxiang

AU - Kim, Junseok

N1 - Funding Information: The first author (C. Lee) was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2019R1A6A3A13094308). 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 thank the editor and the reviewers for their constructive and helpful comments on the revision of this article. Publisher Copyright: © 2021 Global-Science Press.

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N2 - We investigate the effect of cut-off logistic source on evolutionary dynamics of a generalized Cahn-Hilliard (CH) equation in this paper. It is a well-known fact that the maximum principle does not hold for the CH equation. Therefore, a generalized CH equation with logistic source may cause the negative concentration blow-up problem in finite time. To overcome this drawback, we propose the cut-off logistic source such that only the positive value greater than a given critical concentration can grow. We consider the temporal profiles of numerical results in the one-, two-, and three-dimensional spaces to examine the effect of extra mass source. Numerical solutions are obtained using a finite difference multigrid solver. Moreover, we perform numerical tests for tumor growth simulation, which is a typical application of generalized CH equations in biology. We apply the proposed cut-off logistic source term and have good results.

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On the evolutionary dynamics of the cahn-hilliard equation with cut-off mass source

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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