Discrete Data Assimilation in the Lorenz and 2D Navier--Stokes Equations

TL;DR

This paper investigates discrete data assimilation methods for the Lorenz and 2D Navier--Stokes equations, establishing bounds on observation intervals that ensure the approximate solutions converge to the true reference solutions.

Contribution

It provides theoretical bounds on observation timing that guarantee convergence of assimilated solutions for both Lorenz and 2D Navier--Stokes systems.

Findings

Derived bounds on observation intervals for convergence

Proved convergence of assimilated solutions to reference solutions

Applicable to Lorenz and 2D Navier--Stokes equations

Abstract

Consider a continuous dynamical system for which partial information about its current state is observed at a sequence of discrete times. Discrete data assimilation inserts these observational measurements of the reference dynamical system into an approximate solution by means of an impulsive forcing. In this way the approximating solution is coupled to the reference solution at a discrete sequence of points in time. This paper studies discrete data assimilation for the Lorenz equations and the incompressible two-dimensional Navier--Stokes equations. In both cases we obtain bounds on the time interval h between subsequent observations which guarantee the convergence of the approximating solution obtained by discrete data assimilation to the reference solution.