Convergence rates of the front tracking method for conservation laws in the Wasserstein distances

We prove that front tracking approximations to entropy solutions of scalar conservation laws with convex fluxes converge at a rate of $\mathrm{\Delta }{x}^2$ in the 1-Wasserstein distance $W_1$. Assuming positive initial data, we also show that the approximations converge at a rate of $\mathrm{\Delta }x$ in the $\mathrm{\infty }$-Wasserstein distance $W_{\mathrm{\infty }}$. Moreover, from a simple interpolation inequality between $W_1$ and $W_{\mathrm{\infty }}$ we obtain convergence rates in all the $p$-Wasserstein distances: $\mathrm{\Delta }{x}^{1+1\mathrm{/}p}$, $p\in [1,\mathrm{\infty }]$.