After the early work of R. Ni at Bombardier, and the seminal work of P.L Roe, in particular his 1981 JCP paper and its extensions to scalar multidimensional schemes, these schemes can be considered as finite element methods of the streamline diffusion type. The emphasis is put on non-oscillatory properties, in order to be able to compute flow discontinuities, so that they are nonlinear by construction. Indeed shock capturing is done in a totally different manner as for streamline diffusion, allowing for a class of parameter free schemes. In a way, the Residual Distribution methods can be seen as a kind of compromise between high order TVD-like finite difference/finite volume schemes and classical finite element methods, in that they borrow ideas from both communities: geometrical flexibility, the residual concept on one side, and non oscillatory, maximum principle on the other one.

In the talk, we will first consider the case of steady scalar hyperbolic problems, showing how one can systematically construct parameter free essentially non-oscillatory schemes. Then we will move towards steady advection diffusion problems, showing how uniform accuracy, whatever the Peclet/Reynolds number is. The last part of the talk we will consider recent work on unsteady problems. Examples of compressible flows (laminar and turbulent) will be also shown, in order to demonstrate the efficiency of the method, both in accuracy, memory footprint and CPU time.

This is joint work with many colleagues and students among whom Dante de Santis, Mario Ricchiuto, Algiane Froehly, Adam Larat, Mohamed Mezine at INRIA, and many discussions with H. Deconinck (VKI, Belgium) as well as Phil Roe (Michigan, USA). This work has been funded by several EU contracts: the FP6 ADIGMA project (contract AST5-CT-2006-030719), the FP7 IDIHOM project (contract AAT-2010-RTD-1-265780) and the ERC Advanced Grant ADDECCO (contract #226316), as well as a grant of the Swiss National Fund.