Year: 2024
Author: Minghua Chen, Fan Yu, Qingdong Zhang, Zhimin Zhang
Journal of Computational Mathematics, Vol. 42 (2024), Iss. 5 : pp. 1380–1406
Abstract
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.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/jcm.2304-m2022-0140
Journal of Computational Mathematics, Vol. 42 (2024), Iss. 5 : pp. 1380–1406
Published online: 2024-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 27
Keywords: Variable step-size BDF3 method Allen-Cahn equation Spectral norm inequality Stability and convergence analysis.