Data Assimilation algorithm for 3D B\'enard convection in porous media employing only temperature measurements

TL;DR

This paper introduces a continuous data assimilation algorithm for 3D Bénard convection in porous media that uses only temperature measurements to accurately recover the system's state over time.

Contribution

The paper presents a novel nudging-based data assimilation method that guarantees exponential convergence to the true solution using only temperature data.

Findings

Algorithm converges exponentially under proper parameters.

Error estimates are provided for noisy measurements.

Method requires only temperature data, simplifying data collection.

Abstract

In this paper we propose a continuous data assimilation (downscaling) algorithm for the B\'enard convection in porous media using only coarse mesh measurements of the temperature. In this algorithm, we incorporate the observables as a feedback (nudging) term in the evolution equation of the temperature. We show that under an appropriate choice of the nudging parameter and the size of the mesh, and under the assumption that the observed data is error free, the solution of the proposed algorithm converges at an exponential rate, asymptotically in time, to the unique exact unknown reference solution of the original system, associated with the observed (finite dimensional projection of) temperature data. Moreover, we note that in the case where the observational measurements are not error free, one can estimate the error between the solution of the algorithm and the exact reference solution of the system in terms of the error in the measurements.