TY - JOUR
T1 - A Linear Doubly Stabilized Crank-Nicolson Scheme for the Allen–Cahn Equation with a General Mobility
AU - Hou , Dianming
AU - Ju , Lili
AU - Qiao , Zhonghua
JO - Advances in Applied Mathematics and Mechanics
VL - 5
SP - 1009
EP - 1038
PY - 2024
DA - 2024/07
SN - 16
DO - http://doi.org/10.4208/aamm.OA-2023-0067
UR - https://global-sci.org/intro/article_detail/aamm/23284.html
KW - Allen–Cahn equation, general mobility, linear scheme, Crank-Nicolson.
AB - <p style="text-align: justify;">In this paper, a linear second order numerical scheme is developed and investigated for the Allen–Cahn equation with a general positive mobility. In particular,
our fully discrete scheme is mainly constructed based on the Crank-Nicolson formula
for temporal discretization and the central finite difference method for spatial approximation, and two extra stabilizing terms are also introduced for the purpose of improving numerical stability. The proposed scheme is shown to unconditionally preserve
the maximum bound principle (MBP) under mild restrictions on the stabilization parameters, which is of practical importance for achieving good accuracy and stability
simultaneously. With the help of uniform boundedness of the numerical solutions due
to MBP, we then successfully derive $H^1$-norm and $L^∞$-norm error estimates for the
Allen–Cahn equation with a constant and a variable mobility, respectively. Moreover,
the energy stability of the proposed scheme is also obtained in the sense that the discrete free energy is uniformly bounded by the one at the initial time plus a constant.
Finally, some numerical experiments are carried out to verify the theoretical results
and illustrate the performance of the proposed scheme with a time adaptive strategy.</p>