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

TL;DR

This paper introduces a fast, stable, and accurate splitting algorithm for uncertainty quantification in parameterized stochastic Navier-Stokes flow problems, demonstrating superior convergence and performance in high Reynolds number regimes.

Contribution

It develops a novel fully discrete splitting scheme with proven stability and optimal accuracy, integrating it with stochastic collocation for efficient UQ in convection-dominated flows.

Findings

Proven stability of the proposed algorithm.

Demonstrated optimal convergence rates through numerical experiments.

Effective performance in high Reynolds number benchmark problems.

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 ($Re$).