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A Unified \(L^2\) Norm Error Analysis of SAV-BDF Schemes for the Incompressible Navier–Stokes Equations

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Abstract

This article is concerned with the \(L^2\) norm error analysis of high-order BDF methods for the incompressible Navier–Stokes equation subjected to no-slip boundary conditions. To mitigate the complexity of high-order time-step** algorithms caused by decoupling strategies, we combine the prediction–correction technique with a generalized scalar auxiliary variable approach to devise a family of linear, energy stable BDF schemes up to fifth-order accuracy in time. All the proposed schemes can be decoupled into a sequence of Poisson-type equations for the pressure and velocity fields, which significantly reduces the size of the linear systems at each time step. To deal with the essential difficulty raised by the non-A-stability of high-order BDF schemes, we introduce a class of discrete orthogonal convolution kernels to develop a unified framework for the \(L^2\) norm convergence analysis of the proposed high-order BDF schemes. The velocity attains optimal rate of convergence in the \(L^2\) norm under some mild mesh restrictions. Furthermore, building on the recent results concerning the positive definiteness of high-order BDF convolution kernels, we also provide an error estimate of the pressure in a weak \(L^2\) norm. Benchmark examples including the Taylor–Green vortex problem and the regularized lid-driven cavity flow with Reynolds numbers up to 5000 are included to verify the stability and high accuracy of our numerical schemes.

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Acknowledgements

The authors are very grateful to editor, anonymous referees for their invaluable comments and suggestions which have led to improvement of the paper.

Funding

Bingquan Ji is supported by Grants 2022TQ0046 and 2022M720019 from Postdoctoral Science Foundation of China. Hong-lin Liao is supported by a Grant 12071216 from National Natural Science Foundation of China.

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Authors and Affiliations

  1. Institute of Applied Physics and Computational Mathematics, Bei**g, 100193, China

    Bingquan Ji

  2. School of Mathematics, Nan**g University of Aeronautics and Astronautics, Nan**g, 211106, China

    Hong-lin Liao

  3. Key Laboratory of Mathematical Modelling and High Performance Computing of Air Vehicles (NUAA), MIIT, Nan**g, 211106, China

    Hong-lin Liao

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  1. Bingquan Ji
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Correspondence to Hong-lin Liao.

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Ji, B., Liao, Hl. A Unified \(L^2\) Norm Error Analysis of SAV-BDF Schemes for the Incompressible Navier–Stokes Equations. J Sci Comput 100, 5 (2024). https://doi.org/10.1007/s10915-024-02555-9

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  • DOI: https://doi.org/10.1007/s10915-024-02555-9

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