A discrete data assimilation scheme for the solutions of the 2D Navier-Stokes equations and their statistics

TL;DR

This paper develops a discrete data assimilation scheme for 2D Navier-Stokes equations, accommodating realistic measurement scenarios with discretization and errors, and proves exponential convergence and statistical properties.

Contribution

It extends a continuous data assimilation method to discrete measurements with errors, providing convergence guarantees and statistical analysis.

Findings

Exponential convergence of the assimilated solution to the reference.

The scheme works with various observables like Fourier modes.

Provides a statistical framework for the discrete assimilation process.

Abstract

We adapt a previously introduced continuous in time data assimilation (downscaling) algorithm for the 2D Navier-Stokes equations to the more realistic case when the measurements are obtained discretely in time and may be contaminated by systematic errors. Our algorithm is designed to work with a general class of observables, such as low Fourier modes and local spatial averages over finite volume elements. Under suitable conditions on the relaxation (nudging) parameter, the spatial mesh resolution and the time step between successive measurements, we obtain an asymptotic in time estimate of the difference between the approximating solution and the unknown reference solution corresponding to the measurements, in an appropriate norm, which shows exponential convergence up to a term which depends on the size of the errors. A stationary statistical analysis of our discrete data assimilation algorithm is also provided.