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 - <p style="text-align: justify;">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.</p>