TY - JOUR
T1 - Explicit Hybrid Numerical Method for the Allen-Cahn Type Equations on Curved Surfaces
AU - Choi, Yongho
AU - Li, Yibao
AU - Lee, Chaeyoung
AU - Kim, Hyundong
AU - Kim, Junseok
N1 - Funding Information:
Y. Choi was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (NRF-2020R1C1C1A0101153712). Y. Li was supported by the Fundamental Research Funds for Central Universities (XTR 042019005) and the China Postdoctoral Science Foundation (2018M640968). C. Lee was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education (NRF-2019R1A6A3A13094308). H. Kim was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education (NRF-2020R1A6A3A13077105). J. Kim was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education (NRF-2019R1A2C1003053).
Publisher Copyright:
©2021 Global-Science Press
PY - 2021/6
Y1 - 2021/6
N2 - We present a simple and fast explicit hybrid numerical scheme for the motion by mean curvature on curved surfaces in three-dimensional (3D) space. We numerically solve the Allen-Cahn (AC) and conservative Allen-Cahn (CAC) equations on a triangular surface mesh. We use the operator splitting method and an explicit hybrid numerical method. For the AC equation, we solve the diffusion term using a discrete Laplace-Beltrami operator on the triangular surface mesh and solve the reaction term using the closed-form solution, which is obtained using the separation of variables. Next, for the CAC equation, we additionally solve the time-space dependent Lagrange multiplier using an explicit scheme. Our numerical scheme is computationally fast and efficient because we use an explicit hybrid numerical scheme. We perform various numerical experiments to demonstrate the robustness and efficiency of the proposed scheme.
AB - We present a simple and fast explicit hybrid numerical scheme for the motion by mean curvature on curved surfaces in three-dimensional (3D) space. We numerically solve the Allen-Cahn (AC) and conservative Allen-Cahn (CAC) equations on a triangular surface mesh. We use the operator splitting method and an explicit hybrid numerical method. For the AC equation, we solve the diffusion term using a discrete Laplace-Beltrami operator on the triangular surface mesh and solve the reaction term using the closed-form solution, which is obtained using the separation of variables. Next, for the CAC equation, we additionally solve the time-space dependent Lagrange multiplier using an explicit scheme. Our numerical scheme is computationally fast and efficient because we use an explicit hybrid numerical scheme. We perform various numerical experiments to demonstrate the robustness and efficiency of the proposed scheme.
KW - Allen-Cahn equation
KW - Conservative Allen-Cahn equation
KW - Hybrid numerical method
KW - Laplace-Beltrami operator
KW - PDE on surface
KW - Triangular surface mesh
UR - http://www.scopus.com/inward/record.url?scp=85108255887&partnerID=8YFLogxK
U2 - 10.4208/NMTMA.OA-2020-0155
DO - 10.4208/NMTMA.OA-2020-0155
M3 - Article
AN - SCOPUS:85108255887
SN - 1004-8979
VL - 14
SP - 797
EP - 810
JO - Numerical Mathematics
JF - Numerical Mathematics
IS - 3
ER -