Data assimilation for the Navier-Stokes equations using local observables

TL;DR

This paper introduces a local data assimilation algorithm for 2D Navier-Stokes equations, proving its effectiveness for analytic flows and demonstrating high-precision synchronization with minimal, mobile data in numerical tests.

Contribution

It develops and analyzes a novel local observation-based data assimilation method for 2D Navier-Stokes equations, including theoretical guarantees and numerical validation.

Findings

Successful recovery of reference flow within error thresholds.

High-precision synchronization achieved with minimal, mobile data.

Numerical demonstrations confirm effectiveness of the approach.

Abstract

We develop, analyze, and test an approximate, global data assimilation/synchronization algorithm based on purely local observations for the two-dimensional Navier-Stokes equations on the torus. We prove that, for any error threshold, if the reference flow is analytic with sufficiently large analyticity radius, then it can be recovered within that threshold. Numerical computations are included to demonstrate the effectiveness of this approach, as well as variants with data on moving subdomains. In particular, we demonstrate numerically that machine precision synchronization is achieved for mobile data collected from a small fraction of the domain.