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A stable second-order BDF scheme for the three-dimensional Cahn–Hilliard–Hele–Shaw system

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Abstract

We propose a stable scheme to solve numerically the Cahn–Hilliard–Hele–Shaw system in three-dimensional space. In the proposed scheme, we discretize the space and time derivative terms by combining with backward differentiation formula, which turns out to be both second-order accurate in space and time. Using this method, a set of linear elliptic equations are solved instead of the complicated and high-order nonlinear equations. We prove that our proposed scheme is uniquely solvable. We use a linear multigrid solver, which is fast and convergent, to solve the resulting discrete system. The numerical tests indicate that our scheme can use a large time step. The accuracy and other capability of the proposed algorithm are demonstrated by various computational results.

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Acknowledgments

The authors thank the reviewers for the constructive and helpful comments on the revision of this article.

Funding

Y.B. Li is supported by the National Natural Science Foundation of China (No. 11601416) and by the China Postdoctoral Science Foundation (No. 2018M640968). The corresponding author (J.S. Kim) was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2019R1A2C1003053).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, 710049, China

    Yibao Li, Qian Yu & Binhu Xia

  2. Department of Computational Science and Engineering, Yonsei University, Seoul, 03722, Republic of Korea

    Weiwei Fang

  3. Department of Mathematics, Korea University, Seoul, 02841, Republic of Korea

    Junseok Kim

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  1. Yibao Li
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  2. Qian Yu
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  3. Weiwei Fang
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  4. Binhu Xia
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  5. Junseok Kim
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Correspondence to Junseok Kim.

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Communicated by: Long Chen

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Li, Y., Yu, Q., Fang, W. et al. A stable second-order BDF scheme for the three-dimensional Cahn–Hilliard–Hele–Shaw system. Adv Comput Math 47, 3 (2021). https://doi.org/10.1007/s10444-020-09835-6

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