TY - JOUR
T1 - Variable Step-Size BDF3 Method for Allen-Cahn Equation
AU - Chen , Minghua
AU - Yu , Fan
AU - Zhang , Qingdong
AU - Zhang , Zhimin
JO - Journal of Computational Mathematics
VL - 5
SP - 1380
EP - 1406
PY - 2024
DA - 2024/07
SN - 42
DO - http://doi.org/10.4208/jcm.2304-m2022-0140
UR - https://global-sci.org/intro/article_detail/jcm/23282.html
KW - Variable step-size BDF3 method, Allen-Cahn equation, Spectral norm inequality, Stability and convergence analysis.
AB - <p style="text-align: justify;">In this work, we analyze the three-step backward differentiation formula (BDF3) method for solving the Allen-Cahn equation on variable grids. For BDF2 method, the discrete orthogonal convolution (DOC) kernels are positive, the stability and convergence
analysis are well established in [Liao and Zhang, Math. Comp., 90 (2021), 1207–1226]
and [Chen, Yu, and Zhang, arXiv:2108.02910, 2021]. However, the numerical analysis for
BDF3 method with variable steps seems to be highly nontrivial due to the additional degrees of freedom and the non-positivity of DOC kernels. By developing a novel spectral
norm inequality, the unconditional stability and convergence are rigorously proved under
the updated step ratio restriction $r_k :=\tau_k/\tau_{k−1}≤1.405$ for BDF3 method. Finally, numerical experiments are performed to illustrate the theoretical results. To the best of
our knowledge, this is the first theoretical analysis of variable steps BDF3 method for the
Allen-Cahn equation.</p>