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A Modular Voigt Regularization of the Crank-Nicolson Finite Element Method for the Navier-Stokes Equations

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Abstract

We study a modular Crank-Nicolson based Voigt regularization algorithm for the Navier-Stokes equations. This algorithm adds a minimally intrusive module that not only implements Voigt regularization but also adds some numerical dissipation which is not existent in the monolithic algorithms. The additional dissipation induced by the method could act to dampen spurious oscillations, improve stability of numerical solutions, and yield improved accuracy with large-scale dynamics. Within, we prove that the algorithm is unconditionally stable. A convergence analysis is provided whereby \(O(\Delta t^2 + \alpha ^2 + {h^k})\) convergence is proven for velocity solutions. Numerical tests illustrate both the proven stability and convergence properties and the benefit of modular Voigt regularization over the monolithic implementation.

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Data availability

The datasets generated and analysed during the current study are available from the corresponding author on request. The software applications used to generate the results are open source and the corresponding author can provide instructions on how to reproduce the analyses.

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Funding

Y. Rong: Supported by Guangdong Basic and Applied Basic Research Foundation Nos. 2019A1515110503 and National Natural Science Foundation of China Nos. 12001139. J. A. Fiordilino: Supported by NSF grants CBET 1609120 and DMS 1522267 and the DoD SMART Scholarship. F. Shi: Supported by Shenzhen Technology Projects Nos. GXWD20201230155427003-20200822102539001 and RCJC20210609103755110. Y. Cao: Supported by Shenzhen Technology Projects Nos. ZDSYS201707280904031.

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Authors and Affiliations

  1. College of Engineering Physics, Shenzhen Technology University, 518118, Shenzhen, Guangdong, China

    Y. Rong & Y. Cao

  2. Department of Mathematics, University of Pittsburgh, Pittsburgh, PA, 15260, USA

    J. A. Fiordilino

  3. Measurement Science and Engineering Department, Naval Surface Warfare Center Corona Division, Corona, CA, 92878, USA

    J. A. Fiordilino

  4. College of Science, Harbin Institute of Technology, 518055, Shenzhen, Guangdong, China

    F. Shi

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  1. Y. Rong
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Appendix

Appendix

The time-stepping schemes of algorithms used in Sect. 4 are as follows.

CNLE: Find \({\varvec{u}}^{n+1}\) and \(p^{n+1}\) satisfying:

$$\begin{aligned}&\frac{{\varvec{u}}^{n+1}-{\varvec{u}}^{n}}{\Delta t} -\nu \Delta \frac{{\varvec{u}}^{n+1}+{\varvec{u}}^{n}}{2} +\frac{3{\varvec{u}}^{n}-{\varvec{u}}^{n-1}}{2}\cdot \nabla \frac{{\varvec{u}}^{n+1}+{\varvec{u}}^{n}}{2} +\nabla \frac{p^{n+1}+p^{n}}{2} ={\varvec{f}}^{n+\frac{1}{2}}, \\&\quad \nabla \cdot {\varvec{u}}^{n+1}=0. \end{aligned}$$

ModCN: \(Step 1 \). Find \({\varvec{w}}^{n+1}\) and \(\lambda ^{n+1}\) satisfying:

$$\begin{aligned}&\frac{{\varvec{w}}^{n+1}-{\varvec{u}}^{n}}{\Delta t} -\nu \Delta \frac{{\varvec{w}}^{n+1}+{\varvec{u}}^{n}}{2} +\frac{{\varvec{w}}^{n+1}+{\varvec{u}}^{n}}{2}\cdot \nabla \frac{{\varvec{w}}^{n+1}+{\varvec{u}}^{n}}{2} +\nabla \frac{\lambda ^{n+1}+p^{n}}{2} ={\varvec{f}}^{n+\frac{1}{2}}, \\&\quad \nabla \cdot {\varvec{w}}^{n+1}=0. \end{aligned}$$

\(Step 2 \) . Find \({\varvec{u}}^{n+1}\) and \(p^{n+1}\) satisfying:

$$\begin{aligned}&\frac{{\varvec{u}}^{n+1}-{\varvec{w}}^{n+1}}{\Delta t} -\alpha ^{2}\frac{\Delta {\varvec{u}}^{n+1}-\Delta {\varvec{u}}^{n}}{\Delta t} +\nabla \frac{p^{n+1}-\lambda ^{n+1}}{2} =0, \\&\quad \nabla \cdot {\varvec{u}}^{n+1}=0. \end{aligned}$$

MonoCN: Find \({\varvec{u}}^{n+1}\) and \(p^{n+1}\) satisfying:

$$\begin{aligned}&\frac{{\varvec{u}}^{n+1}-{\varvec{u}}^{n}}{\Delta t} -\alpha ^{2}\frac{\Delta {\varvec{u}}^{n+1}-\Delta {\varvec{u}}^{n}}{\Delta t} -\nu \Delta \frac{{\varvec{u}}^{n+1}+{\varvec{u}}^{n}}{2} +\frac{{\varvec{u}}^{n+1}+{\varvec{u}}^{n}}{2}\cdot \nabla \frac{{\varvec{u}}^{n+1}+{\varvec{u}}^{n}}{2} \\&+\nabla \frac{p^{n+1}+p^{n}}{2} ={\varvec{f}}^{n+\frac{1}{2}}, \\&\quad \nabla \cdot {\varvec{u}}^{n+1}=0. \end{aligned}$$

MonoCNLE: Find \({\varvec{u}}^{n+1}\) and \(p^{n+1}\) satisfying:

$$\begin{aligned}&\frac{{\varvec{u}}^{n+1}-{\varvec{u}}^{n}}{\Delta t} -\alpha ^{2}\frac{\Delta {\varvec{u}}^{n+1}-\Delta {\varvec{u}}^{n}}{\Delta t} -\nu \Delta \frac{{\varvec{u}}^{n+1}+{\varvec{u}}^{n}}{2} +\frac{3{\varvec{u}}^{n}-{\varvec{u}}^{n-1}}{2}\cdot \nabla \frac{{\varvec{u}}^{n+1}+{\varvec{u}}^{n}}{2} \\&+\nabla \frac{p^{n+1}+p^{n}}{2} ={\varvec{f}}^{n+\frac{1}{2}}, \\&\quad \nabla \cdot {\varvec{u}}^{n+1}=0. \end{aligned}$$

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Rong, Y., Fiordilino, J.A., Shi, F. et al. A Modular Voigt Regularization of the Crank-Nicolson Finite Element Method for the Navier-Stokes Equations. J Sci Comput 92, 101 (2022). https://doi.org/10.1007/s10915-022-01945-1

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