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
JO - Numerical Mathematics: Theory, Methods and Applications
VL - 3
SP - 797
EP - 810
PY - 2021
DA - 2021/06
SN - 14
DO - http://doi.org/10.4208/nmtma.OA-2020-0155
UR - https://global-sci.org/intro/article_detail/nmtma/19198.html
KW - Allen-Cahn equation, conservative Allen-Cahn equation, Laplace-Beltrami operator,
triangular surface mesh, hybrid numerical method, PDE on surface.
AB - <p style="text-align: justify;">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.</p>