Section: Research Program

Coupling

Analysis and numerical treatment

The coupling of models and numerical codes is an acute problem encountered in practice by many engineers. E. Godlewski and N. Seguin have recently proposed neat techniques for the coupling of hyperbolic systems and numerical codes.

For hyperbolic systems, finite volume methods are often used with explicit time discretization. When the source terms, typically viscosity and friction, have small influence compared to the hyperbolic part, fractional time steps are suitable. This no longer holds when non trivial equilibria between advection and dissipative terms occur and the concept of Asymptotic-Preserving (AP) methods has been proposed to study these difficulties. AP methods make a breakthrough in the numerical resolution of asymptotic perturbations of partial-differential equations.

Another strategy in the quest for a better balance between accuracy and efficiency is the adaptation of models. Indeed, the systems of partial differential equations we consider result from a hierarchy of simplifying assumptions. However, some of these hypotheses may turn out to be unrelevant locally. The adaptation of models thus consists in determining areas where a simplified model (e.g. shallow water type) is valid and where it is not. In the latter case, we may go back to the “parent” model (e.g. Euler) in the corresponding area. This implies to know how to handle the coupling between the aforementioned models from both theoretical and numerical points of view. In particular, the numerical treatment of transmission conditions is a key point.

Coupling problems also arise within the fluid when it contains pollutants, density variations or biological species. In such situations, reaction terms interact with advection effects and need sophisticated treatment for a more complete description.

Data assimilation

Data assimilation consists in a coupling between a model and observation measurements. Developing robust data assimilation methods for hyperbolic-type 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 have recently proposed a new approach based on the kinetic description of the conservation law. Hence, data assimilation is carried out at the kinetic level, using a Luenberger observer. 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 are able to prove the convergence of the model towards the data in case of complete observations in space and time.

This work is done in collaboration with the M3DISIM Inria project-team. M. Doumic and B. Perthame (BANG) also participate.