We explore the behaviour of the implicit centered scheme (which is known to be stable in the L² norm) for hyperbolic equations, in a bounded interval in dimension one, endowed with, for instance, homogeneous Neumann boundary conditions. The numerical solutions show some unexpected periodical in time structures (“ghost solutions”), for general conservation laws from the transport equation to the Euler equations of compressible gas dynamics (both for smooth and discontinuous solutions). In the case of the linear transport equation, we provide of proof of this spectacular phenomenon involving the convergence of a numerical solution toward a very unexpected function. This is a joint work with Hans Henrik Rugh (Université Paris-Sud, Orsay).