Section: New Results

Coupling methods

Data assimilation for conservation laws associated with kinetic description

Participants : Anne-Céline Boulanger, Philippe Moireau [Inria M3DISIM] , Jacques Sainte-Marie.

In order to take advantage of the kinetic description of conservation laws already used for the building of efficient schemes, an innovative data assimilation method for hyperbolic balance laws based in a Luenberger observer on the kinetic equation is developed. It provides a nice theoretical framework for scalar conservation laws, for which we study the cases of complete observations, partial observations in space, in time, and noisy observations. As far as systems are concerned, we focus on the Saint-Venant system, which is hyperbolic, nonlinear and has a topographic source term. We build an observer based only on water depths measurements. Numerical simulations are provided in the case of scalar laws and systems, in one and two dimensions, which validate the efficiency of the method [14] .

Mach-parametrized flows

Participants : Stéphane Dellacherie [CEA] , Bruno Després [UPMC Paris 6] , Yohan Penel.

In order to enrich the modelling of fluid flows, we investigate in this paper a coupling between two models dedicated to distinct regimes. More precisely, we focus on the influence of the Mach number as the low Mach case is known to induce theoretical and numerical issues in a compressible framework. A moving interface is introduced to separate a compressible model (Euler with source term) and its low Mach counterpart through relevant transmission conditions. A global steady state for the coupled problem is exhibited. Numerical simulations are then performed to highlight the influence of the coupling by means of a robust numerical strategy [20] .

Error analysis in a coupling strategy

Participants : Clément Cancès [UPMC Paris 6] , Frédéric Coquel [CNRS] , Edwige Godlewski, Hélène Mathis [Univ. Nantes] , Nicolas Seguin.

We have proposed in a simplified framework an error analysis for an adaptive method which automatically selects the optimal model to use, the choice being between a reference model and an associated simplified one, see [15] . In particular, we are able to balance the thickness of the coupling buffer zone with the threshold on the modelling error which appears when introducing the coarse model.