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Volume 22, Issue 5
The Weighted and Shifted Two-Step BDF Method for Allen-Cahn Equation on Variable Grids

Minghua Chen, Fan Yu & Qingdong Zhang

DOI: 10.4208/ijnam2025-1029

Int. J. Numer. Anal. Mod., 22 (2025), pp. 671-693.

Published online: 2025-05

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  • Abstract

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.

  • Keywords

Weighted and shifted two-step BDF method, variable step size, Allen-Cahn equation, stability and convergence analysis.

  • AMS Subject Headings

65M12, 65M70, 65L06

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-22-671, author = {Chen , MinghuaYu , Fan and Zhang , Qingdong}, title = {The Weighted and Shifted Two-Step BDF Method for Allen-Cahn Equation on Variable Grids }, journal = {International Journal of Numerical Analysis and Modeling}, year = {2025}, volume = {22}, number = {5}, pages = {671--693}, abstract = {

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.

}, issn = {2617-8710}, doi = {https://doi.org/10.4208/ijnam2025-1029}, url = {http://global-sci.org/intro/article_detail/ijnam/24081.html} }
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 -

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.

Chen , MinghuaYu , Fan and Zhang , Qingdong. (2025). The Weighted and Shifted Two-Step BDF Method for Allen-Cahn Equation on Variable Grids . International Journal of Numerical Analysis and Modeling. 22 (5). 671-693. doi:10.4208/ijnam2025-1029
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