The l global decay to discrete shocks for scalar monotone schemes

Given a family of discrete shocks φ of a monotone scheme, we prove that the discrete shock profile with rational shock speed η is asymptotically stable with respect to the l1 topology ‖ · ‖1: if u0 − φ ∈ l1, then ‖un − φ·−nη‖1 → 0 as n → ∞ under no restriction conditions of the initial data to the interval [inf φ, supφ]. The asymptotic wave profile is uniquely identified from the above family by a mass parameter.