  We prove the linear orbital stability of spectrally stable stationary
discrete shock profiles for conservative finite difference schemes applied to
systems of conservation laws. The proof relies on an accurate description of
the pointwise asymptotic behavior of the Green's function associated with those
discrete shock profiles, improving on the result of Lafitte-Godillon [God03].
The main novelty of this stability result is that it applies to a fairly large
family of schemes that introduce some artificial possibly high-order viscosity.
The result is obtained under a sharp spectral assumption rather than by
imposing a smallness assumption on the shock amplitude.
