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

TL;DR

This paper presents RedEigCD, a novel self-adaptive timestepping method for reduced-order models of incompressible flows, which enhances stability and efficiency by leveraging spectral bounds and linear stability theory.

Contribution

The work introduces RedEigCD, the first spectral-based adaptive timestep method for ROMs of Navier-Stokes equations, with proven stability advantages over full-order models.

Findings

RedEigCD achieves up to 40 times larger timesteps than FOMs.

The method maintains accuracy while significantly increasing stability.

The approach links linear stability theory with ROM integration for the first time.

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.