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

Abstract:

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 $\|\cdot \|_1$: if $u^0-\phi \in l^1$, then $\|u^n -\phi _{\cdot - n\eta }\|_1 \to 0$ 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.