Convergence of a mobile data assimilation scheme for the 2D Navier-Stokes equations

TL;DR

This paper proposes a localized, moving observation window data assimilation scheme for the 2D Navier-Stokes equations, demonstrating that fast movement ensures perfect synchronization with the true solution, supported by theoretical proof and numerical simulations.

Contribution

It introduces a novel moving observation window approach for data assimilation in 2D Navier-Stokes, with proven convergence under fast movement and practical strategies for optimal movement.

Findings

Fast-moving observation windows achieve perfect synchronization.

Region-guided movement improves assimilation efficiency.

Numerical results validate theoretical convergence and effectiveness.

Abstract

We introduce a localized version of the nudging data assimilation algorithm for the periodic 2D Navier-Stokes equations in which observations are confined (i.e., localized) to a window that moves across the entire domain along a predetermined path at a given speed. We prove that, if the movement is fast enough, then the algorithm perfectly synchronizes with a reference solution. The analysis suggests an informed scheme in which the subdomain moves according to a region where the error is dominant is optimal. Numerical simulations are presented that compare the efficacy of movement that follows a regular pattern, one guided by the dominant error, and one that is random.