# Continuous Data Assimilation Reduced Order Models of Fluid Flow

## TL;DR

This paper introduces a novel continuous data assimilation reduced order model (DA-ROM) for fluid flow simulation, improving long-term accuracy by integrating measurement data and adaptive nudging strategies, with proven exponential convergence.

## Contribution

The paper combines continuous data assimilation with reduced order models, providing a new approach that enhances long-term accuracy and stability in fluid flow simulations.

## Key findings

- DA-ROM converges exponentially fast to the true solution with proper nudging.
- Adaptive nudging improves long-term accuracy of ROMs.
- Numerical tests confirm the effectiveness of DA-ROM with adaptive strategies.

## Abstract

We propose, analyze, and test a novel continuous data assimilation reduced order model (DA-ROM) for simulating incompressible flows. While ROMs have a long history of success on certain problems with recurring dominant structures, they tend to lose accuracy on more complicated problems and over longer time intervals. Meanwhile, continuous data assimilation (DA) has recently been used to improve accuracy and, in particular, long time accuracy in fluid simulations by incorporating measurement data into the simulation. This paper synthesizes these two ideas, in an attempt to address inaccuracies in ROM by applying DA, especially over long time intervals and when only inaccurate snapshots are available. We prove that with a properly chosen nudging parameter, the proposed DA-ROM algorithm converges exponentially fast in time to the true solution, up to discretization and ROM truncation errors. Finally, we propose a strategy for nudging adaptively in time, by adjusting dissipation arising from the nudging term to better match true solution energy. Numerical tests confirm all results, and show that the DA-ROM strategy with adaptive nudging can be highly effective at providing long time accuracy in ROMs.