Verifiability and Limit Consistency of Eddy Viscosity Large Eddy Simulation Reduced Order Models

TL;DR

The paper tackles the challenge of stable, accurate reduced-order models for under-resolved convection-dominated flows by introducing the Ladyzhenskaya ROM (L-ROM), a generalization of the Smagorinsky ROM (S-ROM). It develops a unified LES-ROM framework and proves verifiability—bounding the ROM error by the closure error—and discrete limit consistency for both L-ROM and S-ROM as the reduced dimension $r$ approaches the snapshot rank $d$ and the ROM lengthscale $δ$ tends to zero. Numerical experiments on the 1D Burgers equation with sharp gradients and the 2D lid-driven cavity at $Re=15{,}000$ confirm the theory, showing that L-ROM can better handle sharp gradients while maintaining stability. The results provide rigorous guarantees for EV-ROM closures and offer practical guidance for deploying LES-ROMs in turbulent flow simulations, with avenues for extending the approach to other closures and filters.

Abstract

Large eddy simulation reduced order models (LES-ROMs) are ROMs that leverage LES ideas (e.g., filtering and closure modeling) to construct accurate and efficient ROMs for convection-dominated (e.g., turbulent) flows. Eddy viscosity (EV) ROMs (e.g., Smagorinsky ROM (S-ROM)) are LES-ROMs whose closure model consists of a diffusion-like operator in which the viscosity depends on the ROM velocity. We propose the Ladyzhenskaya ROM (L-ROM), which is a generalization of the S-ROM. Furthermore, we prove two fundamental numerical analysis results for the new L-ROM and the classical S-ROM: (i) We prove the verifiability of the L-ROM and S-ROM, i.e, that the ROM error is bounded (up to a constant) by the ROM closure error. (ii) We introduce the concept of ROM limit consistency (in a discrete sense), and prove that the L-ROM and S-ROM are limit consistent, i.e., that as the ROM dimension approaches the rank of the snapshot matrix, $d$, and the ROM lengthscale goes to zero, the ROM solution converges to the \emph{``true solution"}, i.e., the solution of the $d$-dimensional ROM. Finally, we illustrate numerically the verifiability and limit consistency of the new L-ROM and S-ROM in two under-resolved convection-dominated problems that display sharp gradients: (i) the 1D Burgers equation with a small diffusion coefficient; and (ii) the 2D lid-driven cavity flow at Reynolds number $Re=15,000$.

Figures (11)

• Figure 1: Schematic of limit consistency in the LES--ROM context.
• Figure 2: The Burgers equation at $\nu = 2 \times 10^{-3}$. Predicted value of $u$ comparison between the G-ROM, S-ROM, and L-ROM at $r = 10$ with $\delta = 0.04$.
• Figure 3: The Burgers equation at $\nu = 2 \times 10^{-3}$. The ROM error $\varepsilon_{\text{ROM}}$ and the closure error $\varepsilon_{\text{closure}}$ with respect to different $r$ values for the S-ROM and L-ROM with $C_S=1$ and $\delta =0.001$.
• Figure 4: The Burgers equation at $\nu = 2 \times 10^{-3}$. The behavior of the ROM error $\varepsilon_{\text{ROM}}$ with respect to the closure error $\varepsilon_{\text{closure}}$. Values shown are for $r = 15, \dots, 35.$
• Figure 5: The Burgers equation at $\nu = 2 \times 10^{-3}$, demonstrating the limit consistency of the S-ROM and L-ROM.

Theorems & Definitions (33)

• Lemma 1: layton2008introductiontemam2001navier
• Lemma 2: Strong monotonicity du1991analysisminty1962monotonelions1969quelques
• Lemma 3: Discrete Gronwall Lemma heywood1990finite
• Remark 1
• Definition 2.1: ROM $L^2$ projection KV01
• Definition 2.2: Generic Constant C
• Lemma 4
• Remark 2
• Remark 3
• Definition 3.1: Verifiability koc2022verifiability