Communications in Mathematical Sciences

Volume 13 (2015)

Number 3

Special Issue in Honor of George Papanicolaou’s 70th Birthday

Guest Editors: Liliana Borcea, Jean-Pierre Fouque, Shi Jin, Lenya Ryzhik, and Jack Xin

Data assimilation for hyperbolic conservation laws: A Luenberger observer approach based on a kinetic description

Pages: 587 – 622

DOI: http://dx.doi.org/10.4310/CMS.2015.v13.n3.a1

Authors

Anne-Celine Boulanger (Laboratoire Jacques-Louis Lions and Inria Paris-Rocquencourt, Paris, France)

Philippe Moireau (Inria Saclay-Île de France, l’Ecole Polytechnique, Palaiseau, France)

Benoît Perthame (Laboratoire Jacques-Louis Lions and Inria Paris-Rocquencourt, Paris, France)

Jacques Sainte-Marie (Inria Paris-Rocquencourt, Compiègne, France)

Abstract

Developing robust data assimilation methods for hyperbolic conservation laws is a challenging subject. Those PDEs indeed show no dissipation effects and the input of additional information in the model equations may introduce errors that propagate and create shocks. We propose a new approach based on the kinetic description of the conservation law. A kinetic equation is a first-order partial differential equation in which the advection velocity is a free variable. In certain cases, it is possible to prove that the nonlinear conservation law is equivalent to a linear kinetic equation. Hence, data assimilation is carried out at the kinetic level, using a Luenberger observer also known as the nudging strategy in data assimilation. Assimilation then resumes to the handling of a BGK type equation. The advantage of this framework is that we deal with a single “linear” equation instead of a nonlinear system and it is easy to recover the macroscopic variables. The study is divided into several steps and essentially based on functional analysis techniques. First, we prove the convergence of the model towards the data in case of complete observations in space and time. Second, we analyze the case of partial and noisy observations. To conclude, we validate our method with numerical results on Burgers equation and emphasize the advantages of this method with the more complex Saint-Venant system.

Keywords

data assimilation, hyperbolic conservation law, kinetic formulation, nudging, shallow water system

2010 Mathematics Subject Classification

35L40, 35L65, 65M08, 76D55, 93E11

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