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Unconditional Stability and Optimal Error Estimates of Euler Implicit/Explicit-SAV Scheme for the Navier–Stokes Equations

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Abstract

The unconditional stability and convergence analysis of the Euler implicit/explicit scheme with finite element discretization are studied for the incompressible time-dependent Navier–Stokes equations based on the scalar auxiliary variable approach. Firstly, a corresponding equivalent system of the Navier–Stokes equations with three variables is formulated, the stable finite element spaces are adopted to approximate these variables and the corresponding theoretical analysis results are provided. Secondly, a fully discrete scheme based on the backward Euler method is developed, the temporal treatment is based on the Euler implicit/explicit scheme, which is implicit for the linear terms and explicit for the nonlinear term. Hence, a constant coefficient algebraic system is formed and it can be solved efficiently. The discrete unconditional energy dissipation and stability of numerical solutions in various norms are established with any restriction on the time step, optimal error estimates are also provided. Finally, some numerical results are provided to illustrate the performances of the considered numerical scheme.

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Data Availability Statement

Raw data were generated at the FreeFEM++ 14.3 64. Derived data supporting the findings of this study are available from the corresponding author upon request.

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Funding

Tong Zhang was supported by the NSF of China (No.11971152) and NSF of Henan Province (202300410167). Jinyun Yuan was supported by the NSF of China (No. 12171087).

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Authors and Affiliations

  1. School of Mathematics and Information Science, Yantai University, Yantai, 264005, People’s Republic of China

    Tong Zhang

  2. School of Computer Science and Technology, Dongguan University of Technology, Dongguan, 523808, People’s Republic of China

    JinYun Yuan

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  1. Tong Zhang
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  2. JinYun Yuan
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Contributions

Tong Zhang carried out the main theorem and wrote the paper, JinYun Yuan revised and checked the paper, Tong Zhang and JinYun Yuan read and approved the final version.

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Correspondence to Tong Zhang.

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Derived data supporting the findings of this study are available from the corresponding author upon request.

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This work was supported by NSF of China (Nos.11971152, 12171087) and NSF of Henan Province (202300410167)).

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Zhang, T., Yuan, J. Unconditional Stability and Optimal Error Estimates of Euler Implicit/Explicit-SAV Scheme for the Navier–Stokes Equations. J Sci Comput 90, 1 (2022). https://doi.org/10.1007/s10915-021-01681-y

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  • DOI: https://doi.org/10.1007/s10915-021-01681-y

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