A Computational Study of a Data Assimilation Algorithm for the Two-dimensional Navier--Stokes Equations

TL;DR

This paper evaluates a data assimilation algorithm for the 2D Navier-Stokes equations, demonstrating its efficiency and accuracy in recovering flow fields from sparse observational data, outperforming theoretical expectations.

Contribution

The study provides the first computational validation showing the algorithm's effectiveness with fewer observations than predicted by theory.

Findings

Nodal observation density needed is lower than analytical estimates.

The method performs comparably to the number of determining Fourier modes.

The algorithm is computationally efficient and robust.

Abstract

We study the numerical performance of a continuous data assimilation (downscaling) algorithm, based on ideas from feedback control theory, in the context of the two-dimensional incompressible Navier--Stokes equations. Our model problem is to recover an unknown reference solution, asymptotically in time, by using continuous-in-time coarse-mesh nodal-point observational measurements of the velocity field of this reference solution (subsampling), as might be measured by an array of weather vane anemometers. Our calculations show that the required nodal observation density is remarkably less that what is suggested by the analytical study; and is in fact comparable to the {\it number of numerically determining Fourier modes}, which was reported in an earlier computational study by the authors. Thus, this method is computationally efficient and performs far better than the analytical estimates suggest.