We present a finite volume scheme, first on the Burgers equations, then on the Euler equations, based on a conservative reconstruction of shocks inside each cells of the mesh. Its main features are the following points. First, the scheme is exact whenever the initial datum is a pure shock, in the sense that the approximate solution is the exact solution averaged over the cells of the mesh. Second, the scheme has in general a very low numerical diffusion and the shocks have a width of one or two cells. Third, no spurious oscillations in the momentum appear behind slowly moving shocks, which is not the case in most of the scheme developed so far. We also present prospective result on the full Euler equations with energy. The wall heating phenomenon, which is an artificial elevation of the temperature when a shock reflects on a wall, is also drastically diminished.
