Data assimilation for hyperbolic conservation laws: a Luenberger observer approach based on a kinetic description (Q2348470)

The authors propose a new approach for the PDE of the hyperbolic conservation laws model based on the kinetic description of the conservation law. 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 one deals 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, one proves the convergence of the model towards the data in case of complete observations in space and time. Second, one analyzes the case of partial and noisy observations. To conclude, the authors validate the method with numerical results on Burgers equation and emphasize the advantages of this method with the more complex Saint-Venant system.