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Second Order Convergence of a Modified MAC Scheme for Stokes Interface Problems

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Abstract

Stokes flow equations have been implemented successfully in practice for simulating problems with moving interfaces. Though computational methods based on the well-known MAC scheme produce accurate solutions and numerical convergence can be demonstrated using a resolution study, the rigorous convergence proofs are usually limited to particular reformulations and boundary conditions. In this paper, a rigorous error analysis of the marker and cell scheme for Stokes interface problems with constant viscosity in the framework of the finite difference method is presented. Without reformulating the problem into elliptic PDEs, the main idea is to use a discrete Ladyzenskaja-Babuska-Brezzi condition and construct auxiliary functions, which satisfy discretized Stokes equations and possess at least second order accuracy in the neighborhood of the moving interface. In particular, the method, for the first time, enables one to prove second order convergence of the velocity gradient in the discrete \(\ell ^2\)-norm, in addition to the velocity and pressure fields. Numerical experiments verify the desired properties of the methods and the expected order of accuracy for both two-dimensional and three-dimensional examples.

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Data Availability

The data that support the findings of this study are available from the corresponding author, upon reasonable request.

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Acknowledgements

H. Dong is partially supported by the NSFC (Grant No. 12001193), by the Scientific Research Fund of Hunan Provincial Education Department (Grant No. 20B376), by the Changsha Municipal Natural Science Foundation (Grant No. kq2014073, kq2208158). W. Ying is partially supported by the NSFC (Grant No. DMS-11771290), by the the Strategic Priority Research Program of Chinese Academy of Sciences (Grant No. XDA25010405), by the Science Challenge Project of China (Grant No. TZ2016002). J. Zhang was partially supported by the National Natural Science Foundation of China (Grant No. 12171376), by the Fundamental Research Funds for the Central Universities (Grant No. 2042021kf0050), and by the Natural Science Foundation of Hubei Province (Grant No. 2019CFA007).

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Authors and Affiliations

  1. MOE-LCSM, School of Mathematics and Statistics, Hunan Normal University, Changsha, 410081, Hunan, People’s Republic of China

    Haixia Dong

  2. School of Mathematical Sciences, and Institute of Natural Sciences, Shanghai Jiao Tong University, Minhang, 200240, Shanghai, People’s Republic of China

    Zhongshu Zhao

  3. Department of Applied Mathematics, Illinois Institute of Technology, Rettaliata Engineering Center, Room 11B, 10 W. 32nd Street, Chicago, IL, 60616, USA

    Shuwang Li

  4. School of Mathematical Sciences, MOE-LSC, and Institute of Natural Sciences, Shanghai Jiao Tong University, Minhang, 200240, Shanghai, People’s Republic of China

    Wenjun Ying

  5. Hubei Key Laboratory of Computational Science, School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China

    Jiwei Zhang

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Haixia Dong is partially supported by the National Natural Science Foundation of China (Grant No. 12001193, the Scientific Research Fund of Hunan Provincial Education Department (No.20B376), Changsha Municipal Natural Science Foundation (No. kq2014073). Wenjun Ying is partially supported by the Strategic Priority Research Program of Chinese Academy of Sciences (Grant No. XDA25010405), the National Natural Science Foundation of China (Grant No. DMS-11771290) and the Science Challenge Project of China (Grant No. TZ2016002).

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Dong, H., Zhao, Z., Li, S. et al. Second Order Convergence of a Modified MAC Scheme for Stokes Interface Problems. J Sci Comput 96, 27 (2023). https://doi.org/10.1007/s10915-023-02239-w

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