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Numerical Integration of Navier-Stokes Equations by Time Series Expansion and Stabilized FEM

Ahmad Deeb, Denys Dutykh

TL;DR

This work addresses stable, high-order time integration of incompressible NS equations by embedding a time-series expansion (TSE) into a FEM framework and stabilizing higher-order terms via a Divergent Series Resummation (DSR) approach. The method converts the nonlinear NS problem into a cascade of linear mixed problems that are solved with Brezzi-stable FEM spaces, and uses Borel–Laplace resummation with a factorial-series (FS) implementation to sum divergent series in time. A stabilization mechanism introducing artificial diffusion is developed to control spurious growth in higher-rank terms, linking to NS$\!$-\alpha-like regularizations and dramatically improving stability and allowable time steps, as demonstrated on Taylor–Green vortex and cylinder-flow tests. The results indicate substantial gains in computational efficiency and stability for high-order TSE-FEM, with clear pathways for extensions to more complex dynamics, variable time stepping, and HPC-scale implementations.

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-Babuska-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.

Numerical Integration of Navier-Stokes Equations by Time Series Expansion and Stabilized FEM

TL;DR

This work addresses stable, high-order time integration of incompressible NS equations by embedding a time-series expansion (TSE) into a FEM framework and stabilizing higher-order terms via a Divergent Series Resummation (DSR) approach. The method converts the nonlinear NS problem into a cascade of linear mixed problems that are solved with Brezzi-stable FEM spaces, and uses Borel–Laplace resummation with a factorial-series (FS) implementation to sum divergent series in time. A stabilization mechanism introducing artificial diffusion is developed to control spurious growth in higher-rank terms, linking to NS-\alpha-like regularizations and dramatically improving stability and allowable time steps, as demonstrated on Taylor–Green vortex and cylinder-flow tests. The results indicate substantial gains in computational efficiency and stability for high-order TSE-FEM, with clear pathways for extensions to more complex dynamics, variable time stepping, and HPC-scale implementations.

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-Babuska-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.
Paper Structure (31 sections, 1 theorem, 111 equations, 22 figures, 3 tables, 1 algorithm)

This paper contains 31 sections, 1 theorem, 111 equations, 22 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. NS equations and TSE technique
  3. Main equation
  4. TSE of the solution
  5. Mixed formulation
  6. FEM framework
  7. Convergence analysis for the Stokes problem
  8. Variational formulation and Brezzi's condition
  9. Error of TSE-FEM approximation
  10. Error Convergence
  11. Discussions
  12. Resummation technique for divergent series
  13. Borel Laplace Resummation
  14. Effective Borel resummation using Factorial series
  15. FS Algorithm
  16. ...and 16 more sections

Key Result

Theorem 1

Let $a(\boldsymbol{u}, \boldsymbol{v})$ be a continuous bilinear form defined on $\mathcal{V} \times \mathcal{V}$, and $b(\boldsymbol{v}, \mathrm{q})$ be a continuous bilinear form defined on $\mathcal{V} \times \mathcal{Q}$. Consider the variational problem for $(\boldsymbol{u}, p) \in \mathcal{V} for $\mathrm{F}$ and $\mathrm{G}$ continuous linear forms on $\mathcal{V}$ and $\mathcal{Q}$ respec

Figures (22)

  • Figure 1: Streamlines of the magnitude of the initial condition vector field of the Taylor-Green vortex.
  • Figure 2: Streamlines of the magnitude of the Taylor-Green's first and second terms approximation in the FEMF by means of the mixed formulation.
  • Figure 3: Error of the approximations of Taylor-Green terms for different mesh size $h$ (left panel) and for different Reynolds numbers (right panel), for a fixed polynomial order $s$.
  • Figure 4: Evolution of the error for the first four terms of the Taylor Green time series solution with respect to the mesh size $h$ with two values of Reynolds number: $\mathrm{Re} = 1$ (left panel) and $\mathrm{Re}=1000$ (right panel).
  • Figure 5: Plot of the error of the TSE-FEM when $h = 1/75$ (left panel) and $h=1/150$ (right panel) and $\Delta \mathrm{t} \rightarrow 0$ and for different truncation rank $\m = {2,3,4,5}$.
  • ...and 17 more figures

Theorems & Definitions (1)

  • Theorem 1