Stable self-adaptive timestepping for Reduced Order Models for incompressible flows

TL;DR

The paper addresses the challenge of stable, efficient time integration for projection-based ROMs of incompressible Navier–Stokes equations. It introduces RedEigCD, a self-adaptive timestepping framework that bounds the ROM stability region using offline-derived eigenbounds of reduced operators and current ROM coefficients, preserving online ROM efficiency. A key theoretical contribution shows that, under linearized assumptions, the maximum stable ROM timestep is at least as large as that of the full-order model, providing a strong stability rationale for ROM acceleration. The method is validated on periodic shear-layer roll-up and non-homogeneous actuator-disk flows, achieving large timestep gains (up to 40x) without compromising accuracy and outperforming Gershgorin-based eigenbound estimates. These results establish a practical, stability-aware time-integration approach for ROMs in incompressible flow simulations, with potential extensions to hyperreduction and broader non-quadratic formulations.

Abstract

This work introduces RedEigCD, the first self-adaptive timestepping technique specifically tailored for reduced-order models (ROMs) of the incompressible Navier-Stokes equations. Building upon linear stability concepts, the method adapts the timestep by directly bounding the stability function of the employed time integration scheme using exact spectral information of matrices related to the reduced operators. Unlike traditional error-based adaptive methods, RedEigCD relies on the eigenbounds of the convective and diffusive ROM operators, whose computation is feasible at reduced scale and fully preserves the online efficiency of the ROM. A central theoretical contribution of this work is the proof, based on the combined theorems of Bendixson and Rao, that, under linearized assumptions, the maximum stable timestep for projection-based ROMs is shown to be larger than or equal to that of their corresponding full-order models (FOMs). Numerical experiments for both periodic and non-homogeneous boundary conditions demonstrate that RedEigCD yields stable timestep increases up to a factor 40 compared to the FOM, without compromising accuracy. The methodology thus establishes a new link between linear stability theory and reduced-order modeling, offering a systematic path towards efficient, self-regulating ROM integration in incompressible flow simulations.

Figures (12)

• Figure 1: Schematic illustration of Bendixson's rectangle (gray) trias_self-adaptive_2011 for the eigenvalues of $F$. $I: \rho(\Omega^{-1}D)$, $II:\rho(\Omega^{-1}C(\mathbf u))$, $III: -\rho(\Omega^{-1}C(\mathbf u))$. The red rectangle represents the stability region bounds. The blue curve represents the boundary of the stability region for a third-order Runge-Kutta scheme.
• Figure 2: Graphical representation of the range of reduced eigenvalues for $i=k$ and $i=k-1$, in red, according to the Poincaré separation theorem.
• Figure 3: Graphical representation of the range of reduced singular values for $i=k$ and $i=k-1$, in red, according to the Poincaré separation theorem for singular values.
• Figure 4: Singular values for the Re=1000 shear-layer roll-up up to $M=200$.
• Figure 5: Timestep ratio between Case B and Case A up to $t=20$ (left) and evolution of the imaginary eigenbound (right) for the roll-up of a shear-layer at $\text{Re}=1000$.

Theorems & Definitions (11)

• Lemma 1
• Lemma 2
• Lemma 3
• proof
• Theorem 1
• proof
• Remark 1: Applicability to time-integration
• Theorem 2
• Theorem 3
• Theorem 4