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Efficient, Accurate, and Robust Penalty-Projection Algorithm for Parameterized Stochastic Navier-Stokes Flow Problems

Neethu Suma Raveendran, Md. Abdul Aziz, Sivaguru S. Ravindran, Muhammad Mohebujjaman

TL;DR

This work addresses uncertainty quantification for parameterized stochastic Navier–Stokes equations in convection-dominated regimes by introducing a fast, robust penalty-projection ensemble framework with grad-div stabilization and ensemble-eddy viscosity (EEV). The Coupled-EEV and the decoupled SPP-EEV schemes share the same system matrix across realizations, enabling efficient fully discrete solves; the SPP-EEV variant is shown to be unconditionally stable and to converge to the Coupled-EEV as the grad-div parameter $\gamma$ increases. The authors couple these time-stepping schemes with stochastic collocation methods on sparse grids, forming SCM-SPP-EEV and SCM-Coupled-EEV to achieve accurate UQ with reduced computational cost. Numerical experiments on Taylor Green–vortex, channel flow over a step, and regularized lid-driven cavity with five-dimensional random viscosity demonstrate optimal spatial and temporal convergence, strong agreement between SCM-SPP-EEV and SCM-Coupled-EEV, and clear advantages of EEV in convection-dominated flows. The results indicate a practical, scalable approach for UQ in complex fluid systems, with potential impact on simulations in engineering and geophysics where high Reynolds numbers and uncertain parameters are common.

Abstract

This paper presents and analyzes a fast, robust, efficient, and optimally accurate fully discrete splitting algorithm for the Uncertainty Quantification (UQ) of parameterized Stochastic Navier-Stokes Equations (SNSEs) flow problems those occur in the convection-dominated regimes. The time-stepping algorithm is an implicit backward-Euler linearized method, grad-div and Ensemble Eddy Viscosity (EEV) regularized, and split using discrete Hodge decomposition. Additionally, the scheme's sub-problems are all designed to have different Right-Hand-Side (RHS) vectors but the same system matrix for all realizations at each time-step. The stability of the algorithm is rigorously proven, and it has been shown that appropriately large grad-div stabilization parameters vanish the splitting error. The proposed UQ algorithm is then combined with the Stochastic Collocation Methods (SCMs). Several numerical experiments are given to verify this superior scheme's predicted convergence rates and performance on benchmark problems for high expected Reynolds numbers ($Re$).

Efficient, Accurate, and Robust Penalty-Projection Algorithm for Parameterized Stochastic Navier-Stokes Flow Problems

TL;DR

This work addresses uncertainty quantification for parameterized stochastic Navier–Stokes equations in convection-dominated regimes by introducing a fast, robust penalty-projection ensemble framework with grad-div stabilization and ensemble-eddy viscosity (EEV). The Coupled-EEV and the decoupled SPP-EEV schemes share the same system matrix across realizations, enabling efficient fully discrete solves; the SPP-EEV variant is shown to be unconditionally stable and to converge to the Coupled-EEV as the grad-div parameter increases. The authors couple these time-stepping schemes with stochastic collocation methods on sparse grids, forming SCM-SPP-EEV and SCM-Coupled-EEV to achieve accurate UQ with reduced computational cost. Numerical experiments on Taylor Green–vortex, channel flow over a step, and regularized lid-driven cavity with five-dimensional random viscosity demonstrate optimal spatial and temporal convergence, strong agreement between SCM-SPP-EEV and SCM-Coupled-EEV, and clear advantages of EEV in convection-dominated flows. The results indicate a practical, scalable approach for UQ in complex fluid systems, with potential impact on simulations in engineering and geophysics where high Reynolds numbers and uncertain parameters are common.

Abstract

This paper presents and analyzes a fast, robust, efficient, and optimally accurate fully discrete splitting algorithm for the Uncertainty Quantification (UQ) of parameterized Stochastic Navier-Stokes Equations (SNSEs) flow problems those occur in the convection-dominated regimes. The time-stepping algorithm is an implicit backward-Euler linearized method, grad-div and Ensemble Eddy Viscosity (EEV) regularized, and split using discrete Hodge decomposition. Additionally, the scheme's sub-problems are all designed to have different Right-Hand-Side (RHS) vectors but the same system matrix for all realizations at each time-step. The stability of the algorithm is rigorously proven, and it has been shown that appropriately large grad-div stabilization parameters vanish the splitting error. The proposed UQ algorithm is then combined with the Stochastic Collocation Methods (SCMs). Several numerical experiments are given to verify this superior scheme's predicted convergence rates and performance on benchmark problems for high expected Reynolds numbers ().

Paper Structure

This paper contains 19 sections, 8 theorems, 113 equations, 4 figures, 4 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Fast, Robust, and Efficient Coupled-EEV algorithm for SNSEs
  4. Fast, Robust, and Efficient SPP-EEV algorithm for SNSEs
  5. Stability Analysis
  6. SCMs
  7. Numerical Experiments
  8. Convergence Rates Verification
  9. SPP-EEV scheme converges to the Coupled-EEV scheme as $\gamma\rightarrow\infty$
  10. Spatial and temporal convergence of the SPP-EEV scheme
  11. Taylor Green-vortex (TGV) Problem taylor1937mechanism
  12. Channel Flow Over a Unit Square Step
  13. Regularized Lid-driven Cavity (RLDC) Problem
  14. Effect of EEV on Convection Dominated Problem
  15. Conclusion
  16. ...and 4 more sections

Key Result

Lemma 2.1

\newlabeldgl Let $\Delta t$, $\mathcal{E}$, $a_n$, $b_n$, $c_n$, $d_n$ be non-negative numbers for $n=1,\cdots, M$ such that then for all $\Delta t> 0,$

Figures (4)

  • Figure 6.1: Variable 5D random viscosity in TGV problem for $\mathbb{E}[\nu]=0.001$: (a) Ensemble average of velocity (shown as speed contour) solution of SCM-SPP-EEV method at $t=1$, and (b) plot of Energy vs. Time for the both SCM-SPP-EEV and SCM-Coupled-EEV methods.
  • Figure 6.2: Variable 5D random viscosity in the flow over a step problem: (a) Ensemble average of velocity solution (shown as streamlines over the speed contour) of SCM-SPP-EEV method at $t=40$, (b) plot of Energy vs. Time for the both SCM-SPP-EEV and SCM-Coupled-EEV methods.
  • Figure 6.3: Variable 5D random viscosity in a RLDC problem with $\mathbb{E}[Re]=15,000$: (a) Velocity solution (shown as streamlines over the speed contour) of SCM-SPP-EEV method at $t=600$, (b) Energy vs. Time plot for both Coupled-EEV, and SCM-SPP-EEV (with $\gamma=$ 1e+4) methods.
  • Figure 6.4: Variable 5D random viscosity in a RLDC problem with $\mathbb{E}[Re]=15,000$: Energy vs. Time as $\mu$ varies. Solution blows up for $\mu=0$.

Theorems & Definitions (22)

  • Lemma 2.1
  • Theorem 3.1
  • proof
  • Remark 3.1
  • Theorem 3.2
  • proof
  • Remark 3.2
  • Lemma 3.3
  • proof
  • Lemma 4.1
  • ...and 12 more