Splitting Method for Stochastic Navier-Stokes Equations

TL;DR

This work tackles the two-dimensional stochastic steady Navier–Stokes equations with additive spatial noise, formulating an operator-splitting approach that separates the dynamics into a deterministic NS component and a stochastic equation. The authors establish the variational framework, prove existence and uniqueness under small-data assumptions, and prove that the splitting solution is equivalent to the original SNS solution, while detailing stability and error analyses. They also introduce a modified splitting format that omits nonlinear terms in the stochastic part, deriving analogous existence, uniqueness, and error bounds, which enable substantial computational savings. Numerical experiments in MATLAB with Taylor–Hood elements and grid-based noise demonstrate near-machine accuracy for the splitting method’s mean solution and confirm significant efficiency gains, supporting the method’s potential for large-scale uncertainty quantification in stochastic flows.

Abstract

This paper investigates the two-dimensional stochastic steady-state Navier-Stokes(NS) equations with additive random noise. We introduce an innovative splitting method that decomposes the stochastic NS equations into a deterministic NS component and a stochastic equation. We rigorously analyze the proposed splitting method from the perspectives of equivalence, stability, existence and uniqueness of the solution. We also propose a modified splitting scheme, which simplified the stochastic equation by omitting its nonlinear terms. A detailed analysis of the solution properties for this modified approach is provided. Additionally, we discuss the statistical errors with both the original splitting format and the modified scheme. Our theoretical and numerical studies demonstrate that the equivalent splitting scheme exhibits significantly enhanced stability compared to the original stochastic NS equations, enabling more effective handling of nonlinear characteristics. Several numerical experiments were performed to compare the statistical errors of the splitting method and the modified splitting method. Notably, the deterministic NS equation in the splitting method does not require repeated solving, and the stochastic equation in the modified scheme is free of nonlinear terms. These features make the modified splitting method particularly advantageous for large-scale computations, as it significantly improves computational efficiency without compromising accuracy.

Figures (3)

• Figure 1: Finite element mesh for domain $\Omega$
• Figure 2: Quiver plot corresponding to velocity field $\bm{\xi}_h,$$\mathbb E[\bm \eta_h], \mathbb E[\bm{\hat{\eta}}_h]$, $\mathbb E_{sh,100}[\textbf{u}], \mathbb E_{mh,100}[\textbf{u}], \mathbb E_{h,100}[\textbf{u}]$ (line 1) and the associated scaled temperature field $|\bm{\xi}_h|,\left|\mathbb E[\bm \eta_h]\right|, \left|\mathbb E[\bm {\hat{\eta}}_h]\right|,$$\left|\mathbb E_{sh,100}[\textbf{u}]\right|$, $\left|\mathbb E_{mh,100}[\textbf{u}]\right|$,$\left|\mathbb E_{h,100}[\textbf{u}]\right|$(line 2) .
• Figure 3: From left to right, there are stochastic solution $\bm{\eta}_h$, modified stochastic solution $\hat{\bm{\eta}}_h$, finite element approximate velocity field based on splitting method $\textbf{u}_{sh}$, finite element approximate velocity field based on modified splitting method $\textbf{u}_{mh}$ and finite element approximate velocity field $\textbf{u}_{h}$. From top to bottom, the scaled temperature fields correspond to $\kappa$=0.2019, 0.5139, 0.8547 and 1.7118 .

Theorems & Definitions (25)

• Remark 2.1: Variations of weak format
• Lemma 2.1: inf-sup condition
• Lemma 2.2: Stability of the solution
• proof
• Theorem 2.3: The existence and uniqueness theorem of the solution for small data
• proof
• Remark 3.1: Variations of weak format
• Theorem 3.1: Equivalence
• proof
• Remark 3.2