The $l^1$ global decay to discrete shocks for scalar monotone schemes

Given a family of discrete shocks $\phi$ of a monotone scheme, we prove that the discrete shock profile with rational shock speed $\eta$ is asymptotically stable with respect to the $l^1$ topology $\Vert\cdot\Vert _1$: if $u^0-\phi \in l^1$, then $\Vert u^n -\phi_{\cdot- n\eta}\Vert _1 \to 0$ as $n\to \infty$ under no restriction conditions of the initial data to the interval $[\inf \phi, \sup \phi]$. The asymptotic wave profile is uniquely identified from the above family by a mass parameter.