## Section: Scientific Foundations

Keywords : unstructured meshes, upwind schemes, complex physical models.

### Numerical schemes for fluid mechanics

A large number of industrial problems can be translated into fluid mechanics ones. They may be coupled with one or more physical models. An example is provided by aeroelastic problems, which have been studied in details by other INRIA teams. Another example is given by flows in pipelines where the fluid (a mixture of air–water–gas) has no very well-understood physical properties. One may also consider problems in aeroacoustics, which become more and more important in everyday life. In some occasions, one needs specific numerical tools because fluids have exotic equation of states, or because the amount of computation becomes huge, as for unsteady flows. Another situation where specific tools are needed is when one is interested in very specific quantities, such as the lift and drag of an airfoil, a situation where commercial tools can only provide a very crude answer.

It is a fact that there are many commercial codes. They allow users to consider some of these examples, but the quality of the solutions is far from being optimal. Moreover, the numerical tools of these codes are often not the most recent ones. An example is the noise generated by vortices crossing through a shock wave. It is, up to our knowledge, even out of reach of the most recent technologies because the numerical resources that would necessitate such simulations are tremendous ! In the same spirit, the simulation of a 3D compressible mixing layer in a complex geometry is also out of reach because very different temporal and physical scales need to be captured, thus we need to invents specific algorithms for that purpose.

In order to reach efficient simulation of complex physical problems, we are working on some fundamental aspects of the numerical analysis of non linear hyperbolic problems. Our goal is to develop schemes that can adapt to the modern computer architectures. More precisely, we are working on a class of numerical schemes specifically tuned for unstructured and hybrid meshes. They have the most possible compact stencil that is compatible with the expected order of accuracy. The order of accuracy typically ranges from two to four. Since the stencil is compact, the implementation on parallel machines becomes simple. The price to pay is that the scheme is necessarily implicit. However, the compactness of the scheme enables to use the high performance parallel linear algebra tools developed by the team for the lowest order version of these schemes. The high order versions of these schemes, that are still under development, will lead to new scientific problems at the border between numerical analysis and computer science. In parallel to these fundemental aspects, we also work on adapting more classical numerical tools to complex physical problems such as those encountered in interface flows, turbulent or multiphase flows.

Within a few years, we expect to be able to consider the physical problems that are now difficult to compute thanks to the know–how coming from our research on compact distribution schemes and the daily discussions with specialists of computer science and scientific computing. These problems range from aeroacoustic to multiphysics problems, such as the one mentionned above.