L1 contraction of a monotone conservative scheme
= L1 contraction of a monotone conservative scheme
{c}
{title2=$\|TU-TV\|_1\leq\|U-V\|_1$}
Let $T$ be a monotone conservative scalar update on a periodic grid. Since $T(U\vee V)\geq TU,TV$, one has $(TU-TV)_+\leq T(U\vee V)-TV$. Conservation makes the sum of the right side equal to $\sum(U-V)_+$. Reverse $U,V$ and add to obtain $\|TU-TV\|_1\leq\|U-V\|_1$. On an infinite grid the same proof holds for summable differences with a Lipschitz flux on the state range.