TY - JOUR
T1 - Numerical Analysis of Crank-Nicolson Scheme for the Allen-Cahn Equation
AU - Chu , Qianqian
AU - Jin , Guanghui
AU - Shen , Jihong
AU - Jin , Yuanfeng
JO - Journal of Computational Mathematics
VL - 5
SP - 655
EP - 665
PY - 2021
DA - 2021/08
SN - 39
DO - http://doi.org/10.4208/jcm.2002-m2019-0213
UR - https://global-sci.org/intro/article_detail/jcm/19381.html
KW - Allen-Cahn Equation, Crank-Nicolson scheme, Maximum principle, Convergence.
AB - <p style="text-align: justify;">We consider numerical methods to solve the Allen-Cahn equation using the second-order Crank-Nicolson scheme in time and the second-order central difference approach in
space. The existence of the finite difference solution is proved with the help of Browder
fixed point theorem. The difference scheme is showed to be unconditionally convergent in $L_∞$ norm by constructing an auxiliary Lipschitz continuous function. Based on this result,
it is demonstrated that the difference scheme preserves the maximum principle without
any restrictions on spatial step size and temporal step size. The numerical experiments
also verify the reliability of the method.</p>