TY - JOUR
T1 - The Weighted and Shifted Two-Step BDF Method for Allen-Cahn Equation on Variable Grids 
AU - Chen , Minghua
AU - Yu , Fan
AU - Zhang , Qingdong
JO - International Journal of Numerical Analysis and Modeling
VL - 5
SP - 671
EP - 693
PY - 2025
DA - 2025/05
SN - 22
DO - http://doi.org/10.4208/ijnam2025-1029
UR - https://global-sci.org/intro/article_detail/ijnam/24081.html
KW - Weighted and shifted two-step BDF method, variable step size, Allen-Cahn equation, stability and convergence analysis.
AB - <p style="text-align: justify;">The weighted and shifted seven-step BDF method is proposed by the authors [Akrivis,
Chen, and Yu, IMA. Numer. Anal., DOI:10.1093/imanum/drae089] for parabolic equation on uniform meshes. In this paper, we study the weighted and shifted two-step BDF method (WSBDF2)
for the Allen-Cahn equation on variable grids. In order to preserve a modified energy dissipation
law at the discrete level, a novel technique is designed to deal with the nonlinear term. The stability and convergence analysis of the WSBDF2 method are rigorously proved by the energy method
under the adjacent time-step ratios $r_s ≥ 4.8645.$ Finally, numerical experiments are implemented
to illustrate the theoretical results. The proposed approach is applicable for the Cahn-Hilliard
equation.</p>