INRIA projectteam : BACCHUS
Leader : Rémi Abgrall
New: European Workshop on High Order Nonlinear Numerical methods for Evolutionary PDES Theory and Applications
HONOM 2013
Bordeaux, March 1822, 2013
The aim of this teamproject is to develop and validate numerical methods adapted to physical problems modeled by a set of partial differential equations having mathematical properties that are, in most of the computational or physical domain, dictated by hyperbolic terms. This type of equations is what denote by essentially hyperbolic PDEs. in the rest of the text, though this wording is quite non standard. A typical example is that of the Navier Stokes equations in fluid dynamics in very high Reynolds: in most of the domain, viscous effect are weak, except near the solid boundaries. Our aim is to make contributions in their numerical approximation from the point of view of accuracy and efficiency so that very large scale computations will become much easier in the coming years.
Our main focus will be on fluid dynamics applications, which are at the core of our know how and of our current research directions, but a priori the techniques developed can be applied to other models having a similar mathematical structure, such as aeroacoustics, geophysics or magnetohydrodynamics (MHD) flows, like in the ITER project, or elastodynamics. Since the partial differential equations (PDE) involved in all these applications have similar properties, its approximation is of similar mathematical nature.
The emergence of new types of massively parallel machines allowing true real size simulations, as well as the increasing demand of the industry, have led today to the following trends in numerical simulation:
 higher accuracy is sought, especially for unsteady problems;
 higher efficiency simulation tools for unsteady problems are or need to be developed;
 an increase in the required level of complexity of the geometry, including a wide range of different length scales (sizes), and of the physical models, eventually including coupling of different physics and multiscale modelling;
 an effort is under way to try to take into account in the simulations the uncertainties in the physical model, or/and the geometry, or/and in other parameters, in order to evaluate an average behavior, variance and other statistical quantities if needed.
This list is certainly not exhaustive. It is essential to have methods which are simple to code and to run on modern high performance computers. The choice we have made here aims at {answering to all these challenges of modern scientific computing: accuracy, simplicity, flexibility.
Our objective is to provide original contributions to the numerical approximations required to tackle the abovementioned algorithmic challenges. We do not expect to bring contributions to the physical modeling (i.e. the step from physics to PDEs and their analysis), nor to the extensive simulations of these problems in order to have a deep understanding of the physics of one given problem or of a class of problems. Once a physical model is set up, our focus will be its numerical approximation, and the way this approximation can be efficiently implemented on modern architectures.
Such a goal implies the collaboration between applied mathematicians and computer scientists. The aim being to develop efficient algorithms, the work to be done cannot be summedup as ``classical'' numerical analysis. Indeed, it will involve contributions of specialists of mesh generation, partitioning and adaption, and specialists of resolution methods, and each step needs to be implemented on modern computer architectures.
A large part of the proposal will be supported by the ADDECCO ERC advanced grant proposal which has been accepted in 2008 and has started in december 2008.
Funding :

Collaborations:
 Stanford University: EA AQUARIUS
 University of Michigan, Ann Arbor
 von Karman Institute (be)
 Politechnico de Milano (It)
 INPG, Grenoble
 ENSAM Paris Tech
 ONERA
 CEA CESTA
 Phimeca
 Incka

