The optimal convergence rate of monotone schemes for conservation laws in the Wasserstein distance

In 1994, Nessyahu, Tadmor and Tassa studied convergence rates of monotone finite volume approximations of conservation laws. For compactly supported, $\Lip^+$-bounded initial data they showed a first-order convergence rate in the Wasserstein distance. Our main result is to prove that this rate is optimal. We further provide numerical evidence indicating that the rate in the case of $\Lip^+$-unbounded initial data is worse than first-order.