TY - JOUR
T1 - An explicit hybrid finite difference scheme for the Allen–Cahn equation
AU - Jeong, Darae
AU - Kim, Junseok
N1 - Funding Information:
The first author (D. Jeong) was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) ( NRF-2017R1E1A1A03070953 ). The corresponding author (Junseok Kim) was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education ( NRF-2016R1D1A1B03933243 ). The authors greatly appreciate the reviewers for their constructive comments and suggestions, which have improved the quality of this paper.
Publisher Copyright:
© 2018 Elsevier B.V.
PY - 2018/10/1
Y1 - 2018/10/1
N2 - In this paper, we propose an explicit hybrid numerical method for solving the Allen–Cahn equation, which models antiphase domain coarsening process in a binary mixture. The proposed method is based on an operator splitting method. First, we solve the linear diffusion part using the explicit Euler method. Second, we solve the nonlinear term using the closed-form analytical solution. We show the stability condition of the proposed numerical scheme. We also show the pointwise boundedness of the numerical solution for the Allen–Cahn equation under a solvability condition. Numerical experiments such as linear stability analysis, traveling wave, motion by mean curvature, image segmentation, and crystal growth are presented to demonstrate the performance of the proposed method.
AB - In this paper, we propose an explicit hybrid numerical method for solving the Allen–Cahn equation, which models antiphase domain coarsening process in a binary mixture. The proposed method is based on an operator splitting method. First, we solve the linear diffusion part using the explicit Euler method. Second, we solve the nonlinear term using the closed-form analytical solution. We show the stability condition of the proposed numerical scheme. We also show the pointwise boundedness of the numerical solution for the Allen–Cahn equation under a solvability condition. Numerical experiments such as linear stability analysis, traveling wave, motion by mean curvature, image segmentation, and crystal growth are presented to demonstrate the performance of the proposed method.
KW - Allen–Cahn equation
KW - Explicit hybrid method
KW - Finite difference method
KW - Operator splitting method
UR - http://www.scopus.com/inward/record.url?scp=85043985209&partnerID=8YFLogxK
U2 - 10.1016/j.cam.2018.02.026
DO - 10.1016/j.cam.2018.02.026
M3 - Article
AN - SCOPUS:85043985209
SN - 0377-0427
VL - 340
SP - 247
EP - 255
JO - Journal of Computational and Applied Mathematics
JF - Journal of Computational and Applied Mathematics
ER -