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A Structure-Preserving, Upwind-SAV Scheme for the Degenerate Cahn–Hilliard Equation with Applications to Simulating Surface Diffusion

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Abstract

This paper establishes a structure-preserving numerical scheme for the Cahn–Hilliard equation with degenerate mobility. First, by applying a finite volume method with upwind numerical fluxes to the degenerate Cahn–Hilliard equation rewritten by the scalar auxiliary variable (SAV) approach, we creatively obtain an unconditionally bound-preserving, energy-stable and fully-discrete scheme, which, for the first time, addresses the boundedness of the classical SAV approach under \(H^{-1}\)-gradient flow. Then, a dimensional-splitting technique is introduced in high-dimensional cases, which greatly reduces the computational complexity while preserves original structural properties. Numerical experiments are presented to verify the bound-preserving and energy-stable properties of the proposed scheme. Finally, by applying the proposed structure-preserving scheme, we numerically demonstrate that surface diffusion is approximated by the Cahn–Hilliard equation with degenerate mobility and Flory–Huggins potential, when the absolute temperature is sufficiently low, which agrees well with the theoretical result by using formal asymptotic analysis.

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The code for the current study is available from the corresponding author on reasonable request.

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Acknowledgements

The numerical calculations in this paper have been done on the supercomputing system in the Supercomputing Center of Wuhan University.

Funding

This work was partially supported by the National Natural Science Foundation of China (Nos. 12001210, 12131010, 12271414, 11871384, 12125103, 12071362, 12301558).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Henan University, Kaifeng, 475004, China

    Qiong-Ao Huang

  2. Center for Applied Mathematics of Henan Province, Henan University, Zhengzhou, 450046, China

    Qiong-Ao Huang

  3. School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China

    Wei Jiang, Jerry Zhijian Yang & Cheng Yuan

  4. Hubei Key Laboratory of Computational Science, Wuhan University, Wuhan, 430072, China

    Wei Jiang & Jerry Zhijian Yang

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  1. Qiong-Ao Huang
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  2. Wei Jiang
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Correspondence to Cheng Yuan.

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Huang, QA., Jiang, W., Yang, J.Z. et al. A Structure-Preserving, Upwind-SAV Scheme for the Degenerate Cahn–Hilliard Equation with Applications to Simulating Surface Diffusion. J Sci Comput 97, 64 (2023). https://doi.org/10.1007/s10915-023-02380-6

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  • DOI: https://doi.org/10.1007/s10915-023-02380-6

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