Workshop on accurate methods for hyperbolic systems :
Residual based schemes, Discontinuous Galerkin methods and adaptationInstitut de Mathématiques, Université Bordeaux I
June 1719, 2002
The numerical simulation of complex flows arising from industrial fluid mechanics problems is generally done via either one of the two classes of methods : finite volume like methods such that the socalled modern upwind high resolution methods, or stabilized finite element methods such as the Galerkin/Least square finite element methods. Among the advantages of the finite volume methods, one can list their geometrical flexibility, their robustness, but the quality of the numerical results may be disappointing when the quality of the mesh is not high enough, in particular when these techniques are employed in conjunction with finite element type meshes. The stabilized finite element schemes like the streamline diffusion method are accurate, in particular for smooth flows, but the stabilization parameters have to be fitted ``manually'' in order to guarantee that the solution does not exhibit spurious oscillations. In between of the finite volume schemes and the stabilized finite element methods, many researchers have shown a growing interest in the socalled Discontinuous Galerkin methods, where the solution is represented by a piecewise continuous polynomial function (as in the finite volume methods), the solution is advanced in time in a similar fashion as for the finite elements methods, and stabilization techniques are borrowed either from one class or the other.Since ten years, another class of scheme has emerged, the socalled upwind residual distributive schemes. The main idea is to construct schemes that are as robustas the finite volume type schemes and as accurate as the stabilized finite element methods by combining constraints coming from the physics of the problem asfor the high order FV methods, and using a continuous representation of the solution as for the Least square methods. Even though still in development, this class ofschemes has reached a good level of maturity. This type of schemes also bear some similarities with some centered residual based schemes recently developped.
All theses schemes can be set naturally in the framework of residual distributive methods. The differences lie in the solution representation (piecewise continuous or globally continuous), the spatial discretisation, and the compactness of the stencil, but in all cases the solution is advanced in time via the signal sent by elements to their vertices.
Each of the schemes are promoted by different scientific communities, and there are few occasion to gather them together.The aim of this workshop is to fill the gap and to provide a state of the art of these techniques by gathering their best specialists. Special interest will be given on the schemes themselves of course, but also on what can be said on a posteriori error estimates, mesh adaptation, the quality of the solution, unsteady flows, industrial flows, MHD problems, etc.
There will be proceedings of the workshop. They will be published in Computer and Fluids .
This workshop is organised within the framework of the RTN HYKE
Schedule



Rémi Abgrall, Mathématiques Appliquées
de Bordeaux, abgrall@math.ubordeaux.fr


Institut de Mathématiques, Université
Bordeaux I

updated May 24 2002, abgrall@math.ubordeaux.fr