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Optimized Ventcel-Schwarz methods for the Cahn-Hilliard equation discretized by the stabilized linear Crank-Nicolson scheme

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Abstract

The stabilized linear Crank-Nicolson (SL-CN) scheme is a very important time discretization for the Cahn-Hilliard (CH) equation since it is an unconditionally energy stable method of second order, and allows to use time steps as large as possible to reduce the total calculation. Though, it still requires a very large amount of calculations for simulating the CH equation because of the essential nature of the CH equation. To accelerate the simulation process, we propose in this paper to solve the differential system resulting from the time discretization by an optimized Schwarz method using a newly proposed Ventcel transmission condition. For a setting of two-subdomain domain decomposition with or without overlap, we derive using Fourier analysis the convergence factor, which takes on two different forms according to the size of the time steps. By solving the hard min-max problem of convergence factors using asymptotic analysis, we rigorously optimized the convergence factors for each case and obtained the optimized transmission parameters in explicit form, and the estimate for the corresponding convergence rates. The theoretical results are illustrated by several numerical examples.

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Acknowledgements

The authors are grateful to the anonymous referees for their valuable comments that improved the presentation greatly.

Funding

This work was funded by the National Key R&D Program of China (No. 2021YFA1003400), NSFC-12071069, the Science and Technology Development Planning of Jilin Province (No. YDZJ202201ZYTS573), the Fundamental Research Funds for the Central Universities (No. JGPY202101), and the Space Optoelectronic Measurement & Perception Laboratory of BICE (No. LabSOMP-2021-04).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Northeast Normal University, 5268 Renmin Street, Changchun, 130024, Jilin Province, People’s Republic of China

    Yafei Sun, Yingxiang Xu, Shuangbin Wang & Shan Gao

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  1. Yafei Sun
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  2. Yingxiang Xu
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  3. Shuangbin Wang
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  4. Shan Gao
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Contributions

Yafei Sun and Yingxiang Xu wrote the article; Yingxiang Xu, Shuangbin Wang and Shan Gao developed the theory; and Yafei Sun performed the numerical experiments.

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Correspondence to Yingxiang Xu.

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Communicated by Francesca Rapetti.

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Sun, Y., Xu, Y., Wang, S. et al. Optimized Ventcel-Schwarz methods for the Cahn-Hilliard equation discretized by the stabilized linear Crank-Nicolson scheme. Adv Comput Math 48, 67 (2022). https://doi.org/10.1007/s10444-022-09982-y

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