Continuous data assimilation algorithm for simplified Bardina model

Débora A F Albanez,Maicon J Benvenutti

doi:10.3934/eect.2018002

Abstract

We present a continuous data assimilation algorithm for three-dimensional viscous simplified Bardina turbulence model, based on the fact that dissipative dynamical systems possess finite degrees of freedom. We construct an approximating solution of simplified Barbina model through an interpolant operator which is obtained using observational data of the system. This interpolant is inserted to theoric model coupled to a relaxation parameter, and our main result provides conditions on the finite-dimensional spatial resolution of collected measurements sufficient to ensure that the approximating solution converges to the theoric solution of the model. Global well-posedness of approximating solutions and related results with degrees of freedom are also presented.

Highlights

Data analysis is a wide range of techniques that conciliates mathematical models and physical observations with the goal of optimizing forecasts in evolutionary phenomenons
In the context of Navier-Stokes-α system, it seems to be unknown if the algorithm is convergent without diffusion in the nudging. We prove that it is true for a related system called three-dimensional viscous simplified Bardina turbulence model
We present here three examples of such interpolants

Summary

Introduction

Data analysis is a wide range of techniques that conciliates mathematical models and physical observations with the goal of optimizing forecasts in evolutionary phenomenons. Due to the high degree of freedom of many models, a hard obstacle is to predict suitable initialization points, which in general are deduced starting from discrete and heterogeneous grids of collected observations Approaches to this problem have been topic of countless theoretical and numerical studies. In [16], authors performed numerical studies with the shallow water equations comparing methods involving direct insertion, data assimilation having only Newtonian relaxation and data assimilation including diffusion relaxation. For distinct experiments, they obtained different results and concluded that further inspections are required to determine the practical efficacy of each approach (see [39]).

The following continuous data assimilation is considered

We have