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Volume 42, Issue 5
Variable Step-Size BDF3 Method for Allen-Cahn Equation

Minghua Chen, Fan Yu, Qingdong Zhang & Zhimin Zhang

DOI: 10.4208/jcm.2304-m2022-0140

J. Comp. Math., 42 (2024), pp. 1380-1406.

Published online: 2024-07

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  • 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.

  • Keywords

Variable step-size BDF3 method, Allen-Cahn equation, Spectral norm inequality, Stability and convergence analysis.

  • AMS Subject Headings

65L06, 65M12

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-42-1380, author = {Chen , MinghuaYu , FanZhang , Qingdong and Zhang , Zhimin}, title = {Variable Step-Size BDF3 Method for Allen-Cahn Equation}, journal = {Journal of Computational Mathematics}, year = {2024}, volume = {42}, number = {5}, pages = {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.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2304-m2022-0140}, url = {http://global-sci.org/intro/article_detail/jcm/23282.html} }
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 -

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.

Minghua Chen, Fan Yu, Qingdong Zhang & Zhimin Zhang. (2024). Variable Step-Size BDF3 Method for Allen-Cahn Equation. Journal of Computational Mathematics. 42 (5). 1380-1406. doi:10.4208/jcm.2304-m2022-0140
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