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Numerical integration of Navier–Stokes equations by time series expansion and stabilized FEM

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  • Deeb, Ahmad
  • Dutykh, Denys

Abstract

This manuscript introduces an advanced numerical approach for the integration of incompressible Navier-Stokes (NS) equations using a Time Series Expansion (TSE) method within a Finite Element Method (FEM) framework. The technique is enhanced by a novel stabilization strategy, incorporating a Divergent Series Resummation (DSR) technique, which significantly augments the computational efficiency of the algorithm. The stabilization mechanism is meticulously designed to improve the stability and validity of computed series terms, enabling the application of the Factorial Series (FS) algorithm for series resummation. This approach is pivotal in addressing the challenges associated with the accurate and stable numerical solution of NS equations, which are critical in Computational Fluid Dynamics (CFD) applications. The manuscript elaborates on the variational formulation of Stokes problem and present convergence analysis of the method using the Ladyzhenskaya–Babuvska–Brezzi (LBB) condition. It is followed by the NS equations and the implementation details of the stabilization technique, underscored by numerical tests on laminar flow past a cylinder, showcasing the method’s efficacy and potential for broad applicability in fluid dynamics simulations. The results of the stabilization indicate a substantial enhancement in computational stability and accuracy, offering a promising avenue for future research in the field.

Suggested Citation

  • Deeb, Ahmad & Dutykh, Denys, 2025. "Numerical integration of Navier–Stokes equations by time series expansion and stabilized FEM," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 233(C), pages 208-236.
    • Handle: RePEc:eee:matcom:v:233:y:2025:i:c:p:208-236
    DOI: 10.1016/j.matcom.2025.01.023
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    References listed on IDEAS

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    1. Deeb, Ahmad & Hamdouni, Aziz & Razafindralandy, Dina, 2022. "Performance of Borel–Padé–Laplace integrator for the solution of stiff and non-stiff problems," Applied Mathematics and Computation, Elsevier, vol. 426(C).
    2. Zheng, Bo & Shang, Yueqiang, 2020. "Local and parallel stabilized finite element algorithms based on the lowest equal-order elements for the steady Navier–Stokes equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 178(C), pages 464-484.
    3. Cooke, Charlie H. & Blanchard, Doris K., 1980. "A higher order finite element algorithm for the unsteady Navier-Stokes equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 22(2), pages 127-132.
    4. Zhu, Liping & Chen, Zhangxin, 2015. "A two-level stabilized nonconforming finite element method for the stationary Navier–Stokes equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 114(C), pages 37-48.
    5. Eymard, Robert & Feron, Pierre & Guichard, Cindy, 2018. "Family of convergent numerical schemes for the incompressible Navier–Stokes equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 144(C), pages 196-218.
    6. Zhang, Tong & He, Yinnian, 2012. "Fully discrete finite element method based on pressure stabilization for the transient Stokes equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 82(8), pages 1496-1515.
    7. Breckling, Sean & Fiordilino, Joseph & Reyes, Jorge & Shields, Sidney, 2024. "A note on the long-time stability of pressure solutions to the 2D Navier Stokes equations," Applied Mathematics and Computation, Elsevier, vol. 478(C).
    8. Anselmann, Mathias & Bause, Markus, 2021. "Higher order Galerkin–collocation time discretization with Nitsche’s method for the Navier–Stokes equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 189(C), pages 141-162.
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