Convergence rates of the front tracking method for conservation laws in the Wasserstein distances
Provides rigorous convergence rates for a numerical method in Wasserstein distances, addressing a gap in the analysis of conservation laws for practitioners using optimal transport metrics.
The paper proves that front tracking approximations to entropy solutions of scalar conservation laws with convex fluxes converge at a rate of Δx² in the 1-Wasserstein distance and Δx in the ∞-Wasserstein distance, with rates in all p-Wasserstein distances derived via interpolation.
We prove that front tracking approximations to entropy solutions of scalar conservation laws with convex fluxes converge at a rate of $Δx^2$ in the 1-Wasserstein distance $W_1$. Assuming positive initial data, we also show that the approximations converge at a rate of $Δx$ 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: $Δx^{1+1/p}$, $p \in [1,\infty]$.