Comptes Rendus
Data assimilation and pollution forecasting in Burgers' equation with model error function
Comptes Rendus. Mécanique, Volume 347 (2019) no. 5, pp. 423-444.

This article presents a correction method for a better resolution of the problem of estimating and predicting pollution, governed by Burgers' equations. The originality of the method consists in the introduction of an error function into the system's equations of state to model uncertainty in the model. The initial conditions and diffusion coefficients, present in the equations for pollution and concentration, and also those in the model error equations, are estimated by solving a data assimilation problem. The efficiency of the correction method is compared with that produced by the traditional method without introduction of an error function.

Three test cases are presented in this study in order to compare the performances of the proposed methods. In the first two tests, the reference is the analytical solution and the last test is formulated as part of the “twin experiment”.

The numerical results obtained confirm the important role of the model error equation for improving the prediction capability of the system, in terms of both accuracy and speed of convergence.

Cet article présente une méthode de correction permettant de mieux résoudre le problème de l'estimation et de la prédiction de la pollution décrit par les équations de Burgers. L'originalité de la méthode consiste en l'introduction d'une fonction d'erreur dans le modèle d'état du système pour modéliser l'incertitude du modèle initial. Les conditions initiales et les coefficients de diffusion, présents dans les équations de pollution et de concentration, ainsi que ceux des équations d'erreur du modèle, sont estimés en résolvant un problème d'assimilation de données. L'efficacité de la méthode de correction est comparée à celle offerte par la méthode traditionnelle sans introduction d'une fonction d'erreur.

Trois cas de tests sont présentés dans cette étude pour comparer les performances des méthodes utilisées. Dans les deux premiers tests, la référence est la solution analytique, et le dernier test est formulé dans le cadre de « l'expérience jumelle ». Les résultats numériques obtenus confirment le rôle important de l'équation d'erreur du modèle dans l'amélioration de la capacité de prédiction du système, en termes de précision et de rapidité de convergence de la méthode de correction.

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Published online:
DOI: 10.1016/j.crme.2019.02.002
Keywords: Burgers' equation, Water pollution, Data assimilation, Optimal method BFGS
Mot clés : Équation de Burgers, Pollution de l'eau, Assimilation de données, Méthode optimale BFGS

Tran Thu Ha 1; Nguyen Hong Phong 1; François-Xavier Le Dimet 2; Hong Son Hoang 3

1 Institute of Mechanics, 264 Doi Can, Graduate University of Science and Technology, Vietnam Academy of Science and Technology (VAST), 18 Hoang Quoc Viet, University of Engineering and Technology, VNU, 144 Xuan Thuy, Hanoï, Viet Nam
2 Laboratoire Jean-Kuntzmann, 51, rue des Maths, 38400 Saint-Martin-d'Hères, France
3 REC/HOM/SHOM, 42, avenue Gaspard-Coriolis, 31000 Toulouse, France
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Tran Thu Ha; Nguyen Hong Phong; François-Xavier Le Dimet; Hong Son Hoang. Data assimilation and pollution forecasting in Burgers' equation with model error function. Comptes Rendus. Mécanique, Volume 347 (2019) no. 5, pp. 423-444. doi : 10.1016/j.crme.2019.02.002. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2019.02.002/

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