  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.
