Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Two Robust, Efficient, and optimally Accurate Algorithms for parameterized stochastic navier-stokes Flow Problems

Neethu Suma Raveendran, Md Abdul Aziz, Muhammad Mohebujjaman

Abstract

This paper presents and analyzes two robust, efficient, and optimally accurate fully discrete finite element algorithms for computing the parameterized Navier-Stokes Equations (NSEs) flow ensemble. The timestepping algorithms are linearized, use the backward-Euler method for approximating the temporal derivative, and Ensemble Eddy Viscosity (EEV) regularized. The first algorithm is a coupled ensemble scheme, and the second algorithm is decoupled using projection splitting with grad-div stabilization. We proved the stability and convergence theorems for both algorithms. We have shown that for sufficiently large grad-div stabilization parameters, the outcomes of the projection scheme converge to the outcomes of the coupled scheme. We then combine the Stochastic Collocation Methods (SCMs) with the proposed two Uncertainty Quantification (UQ) algorithms. A series of numerical experiments are given to verify the predicted convergence rates and performance of the schemes on benchmark problems, which shows the superiority of the splitting algorithm.

Two Robust, Efficient, and optimally Accurate Algorithms for parameterized stochastic navier-stokes Flow Problems

Abstract

This paper presents and analyzes two robust, efficient, and optimally accurate fully discrete finite element algorithms for computing the parameterized Navier-Stokes Equations (NSEs) flow ensemble. The timestepping algorithms are linearized, use the backward-Euler method for approximating the temporal derivative, and Ensemble Eddy Viscosity (EEV) regularized. The first algorithm is a coupled ensemble scheme, and the second algorithm is decoupled using projection splitting with grad-div stabilization. We proved the stability and convergence theorems for both algorithms. We have shown that for sufficiently large grad-div stabilization parameters, the outcomes of the projection scheme converge to the outcomes of the coupled scheme. We then combine the Stochastic Collocation Methods (SCMs) with the proposed two Uncertainty Quantification (UQ) algorithms. A series of numerical experiments are given to verify the predicted convergence rates and performance of the schemes on benchmark problems, which shows the superiority of the splitting algorithm.

Paper Structure

This paper contains 17 sections, 8 theorems, 140 equations, 4 figures, 3 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Efficient Coupled EEV (Coupled-EEV) algorithm for SNSEs
  4. Efficient Stabilized Penalty Projection EEV (SPP-EEV) algorithm for SNSEs
  5. Stability Analysis
  6. SCM
  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 2 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 a 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 (18)

  • Lemma 2.1
  • Theorem 3.1: Stability
  • proof
  • Theorem 3.2: Convergence
  • proof
  • Lemma 3.3
  • proof
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • ...and 8 more