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Numerical Analysis of a Convex-Splitting BDF2 Method with Variable Time-Steps for the Cahn–Hilliard Model

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Abstract

In this paper, the convex-splitting BDF2 method with variable time-steps (proposed in Chen et al. SIAM J Numer Anal 57:495–525, 2019) is reconsidered for the Cahn–Hilliard model. We adopt the Fourier pseudo-spectral discretization in space. With the help of the discrete gradient structure of the BDF2 formula and some embedded inequalities, we prove that the scheme preserves a modified energy dissipation law under the updated time-step-ratio restriction \(0<r_{k}=\tau _{k}/\tau _{k-1}<4.864\). By utilizing the discrete orthogonal convolution kernels, some discrete convolution inequalities and some proof techniques (Lemma 4.6), we tackle the difficulty brought from the Douglas-Dupont regularization stabilized term and prove the robust \(L^{2}\) norm convergence of the scheme under the same mild time-step-ratio restriction \(0<r_{k}<4.864.\) Numerical experiments are carried out to support our theoretical analysis and a time adaptive strategy is applied to accelerate the simulation of the multi-scale characteristics of the Cahn–Hilliard model.

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Funding

Xiuling Hu is supported by the National Science Foundation of China under Grant nos. 11501262,12271467.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, 221116, China

    Xiuling Hu

  2. School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, 221116, China

    Lulu Cheng

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  1. Xiuling Hu
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  2. Lulu Cheng
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Contributions

All authors contributed to the manuscript. Numerical analysis and experiments were performed by XH and LC. The first draft of the manuscript was written by XH and all authors checked on previous versions of the manuscript. All authors read and approved the final manuscript.

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Correspondence to Xiuling Hu.

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Hu, X., Cheng, L. Numerical Analysis of a Convex-Splitting BDF2 Method with Variable Time-Steps for the Cahn–Hilliard Model. J Sci Comput 98, 18 (2024). https://doi.org/10.1007/s10915-023-02400-5

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  • DOI: https://doi.org/10.1007/s10915-023-02400-5

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