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

TL;DR

This paper establishes convergence rates for front tracking methods solving scalar conservation laws in various Wasserstein distances, showing quadratic convergence in $W_1$ and linear in $W_ty$, with intermediate rates for all $p$-Wasserstein distances.

Contribution

The paper provides new theoretical convergence rate results for front tracking approximations in Wasserstein distances, including explicit rates in $W_1$, $W_ty$, and all intermediate $p$-Wasserstein metrics.

Findings

Convergence rate of $x^2$ in $W_1$ distance.

Convergence rate of $x$ in $W_ty$ distance.

Interpolated convergence rates for all $p$-Wasserstein distances.

Abstract

We prove that front tracking approximations to entropy solutions of 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: $\Delta x^{1+1/p}$, $p \in [1,\infty]$.