Convergence Rates of the Front Tracking Method for Conservation Laws in the Wasserstein Distances

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