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Double stabilizations and convergence analysis of a second-order linear numerical scheme for the nonlocal Cahn-Hilliard equation

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Abstract

In this paper, we study a second-order accurate and linear numerical scheme for the nonlocal Cahn-Hilliard equation. The scheme is established by combining a modified Crank-Nicolson approximation and the Adams-Bashforth extrapolation for the temporal discretization, and by applying the Fourier spectral collocation to the spatial discretization. In addition, two stabilization terms in different forms are added for the sake of the numerical stability. We conduct a complete convergence analysis by using the higher-order consistency estimate for the numerical scheme, combined with the rough error estimate and the refined estimate. By regarding the numerical solution as a small perturbation of the exact solution, we are able to justify the discrete ℓ∞ bound of the numerical solution, as a result of the rough error estimate. Subsequently, the refined error estimate is derived to obtain the optimal rate of convergence, following the established ℓ∞ bound of the numerical solution. Moreover, the energy stability is also rigorously proved with respect to a modified energy. The proposed scheme can be viewed as the generalization of the second-order scheme presented in an earlier work, and the energy stability estimate has greatly improved the corresponding result therein.

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Acknowledgements

This work was supported by the Chinese Academy of Sciences (CAS) Academy of Mathematics and Systems Science (AMSS) and the Hong Kong Polytechnic University (PolyU) Joint Laboratory of Applied Mathematics. The first author was supported by the Hong Kong Research Council General Research Fund (Grant No. 15300821) and the Hong Kong Polytechnic University Grants (Grant Nos. 1-BD8N, 4-ZZMK and 1-ZVWW). The second author was supported by the Hong Kong Research Council Research Fellow Scheme (Grant No. RFS2021-5S03) and General Research Fund (Grant No. 15302919). The third author was supported by US National Science Foundation (Grant No. DMS-2012269).

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Authors and Affiliations

  1. Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, China

    Xiao Li & Zhonghua Qiao

  2. Department of Mathematics, The University of Massachusetts, North Dartmouth, MA, 02747, USA

    Cheng Wang

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  1. Xiao Li
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  2. Zhonghua Qiao
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  3. Cheng Wang
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Correspondence to Zhonghua Qiao.

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Li, X., Qiao, Z. & Wang, C. Double stabilizations and convergence analysis of a second-order linear numerical scheme for the nonlocal Cahn-Hilliard equation. Sci. China Math. 67, 187–210 (2024). https://doi.org/10.1007/s11425-022-2036-8

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