Séminaire
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 restrict the performance ofa numerical prediction.We propose an approach based on the kinetic description of conservation laws. 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.We consider in this work a general class of conservation laws that may contain non-linearities as typically illustrated by transport equations, Burgers' equation, the shallow water or Euler systems and we focus especially on the non-viscous configurations.During this presentation, we first describe the nudging technique at the kinetic level and the obtained convergence results. Then we propose numerical results including analytical validations for the shallow water equations and related models.
A.-C. Boulanger, P. Moireau, B. Perthame, J. Sainte-Marie. Data Assimilation for hyperbolic conservation laws. A Luenberger observer approach based on a kinetic description. Submitted, ArXiv e-prints, 2013.
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