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Volume 14, Issue 1
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

Numer. Math. Theor. Meth. Appl., 14 (2021), pp. 242-260.

Published online: 2020-10

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  • 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.

  • Keywords

Cahn-Hilliard equation, logistic source, finite difference method, tumor growth application.

  • AMS Subject Headings

65M06, 65M55, 65Z05, 68U20

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-14-242, author = {Lee , ChaeyoungKim , HyundongYoon , SunghaPark , JintaeKim , SangkwonYang , Junxiang and Kim , Junseok}, title = {On the Evolutionary Dynamics of the Cahn-Hilliard Equation with Cut-Off Mass Source}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2020}, volume = {14}, number = {1}, pages = {242--260}, 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.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2020-0051}, url = {http://global-sci.org/intro/article_detail/nmtma/18334.html} }
TY - JOUR T1 - On the Evolutionary Dynamics of the Cahn-Hilliard Equation with Cut-Off Mass Source AU - Lee , Chaeyoung AU - Kim , Hyundong AU - Yoon , Sungha AU - Park , Jintae AU - Kim , Sangkwon AU - Yang , Junxiang AU - Kim , Junseok JO - Numerical Mathematics: Theory, Methods and Applications VL - 1 SP - 242 EP - 260 PY - 2020 DA - 2020/10 SN - 14 DO - http://doi.org/10.4208/nmtma.OA-2020-0051 UR - https://global-sci.org/intro/article_detail/nmtma/18334.html KW - Cahn-Hilliard equation, logistic source, finite difference method, tumor growth application. AB -

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.

Lee , ChaeyoungKim , HyundongYoon , SunghaPark , JintaeKim , SangkwonYang , Junxiang and Kim , Junseok. (2020). On the Evolutionary Dynamics of the Cahn-Hilliard Equation with Cut-Off Mass Source. Numerical Mathematics: Theory, Methods and Applications. 14 (1). 242-260. doi:10.4208/nmtma.OA-2020-0051
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