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Arbitrarily High Order and Fully Discrete Extrapolated RK–SAV/DG Schemes for Phase-field Gradient Flows

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Abstract

In this paper, we construct and analyze a fully discrete method for phase-field gradient flows, which uses extrapolated Runge–Kutta with scalar auxiliary variable (RK–SAV) method in time and discontinuous Galerkin (DG) method in space. We propose a novel technique to decouple the system, after which only several elliptic scalar problems with constant coefficients need to be solved independently. Discrete energy decay property of the method is proved for gradient flows. The scheme can be of arbitrarily high order both in time and space, which is demonstrated rigorously for the Allen–Cahn equation and the Cahn–Hilliard equation. More precisely, optimal \(L^2\)-error bound in space and qth-order convergence rate in time are obtained for q-stage extrapolated RK–SAV/DG method. Several numerical experiments are carried out to verify the theoretical results.

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Acknowledgements

This work is partially supported by the National Science Foundation of China and Hong Kong RGC Joint Research Scheme (NSFC/RGC 11961160718), and the fund of the Guangdong Provincial Key Laboratory of Computational Science and Material Design (No. 2019B030301001). The work of J. Yang is supported by the National Science Foundation of China (NSFC-11871264) and the Shenzhen Natural Science Fund (RCJC20210609103819018).

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Authors and Affiliations

  1. Division of Science and Technology, BNU-HKBU United International College, Zhuhai, Guangdong, China

    Tao Tang

  2. SUSTech International Center for Mathematics, Southern University of Science and Technology, Shenzhen, China

    Tao Tang

  3. School of Mathematics, Harbin Institute of Technology, Harbin, China

    Xu Wu

  4. Department of Mathematics, Southern University of Science and Technology, Shenzhen, China

    Xu Wu & Jiang Yang

  5. Guangdong Provincial Key Laboratory of Computational Science and Material Design, Southern University of Science and Technology, Shenzhen, China

    Jiang Yang

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  1. Tao Tang
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Correspondence to Xu Wu.

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Tang, T., Wu, X. & Yang, J. Arbitrarily High Order and Fully Discrete Extrapolated RK–SAV/DG Schemes for Phase-field Gradient Flows. J Sci Comput 93, 38 (2022). https://doi.org/10.1007/s10915-022-01995-5

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  • DOI: https://doi.org/10.1007/s10915-022-01995-5

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