Susanne Solem

Ulrik S. Fjordholm, Susanne Solem

We prove that a class of monotone, \emph{$W_1$-contractive} schemes for scalar conservation laws converge at a rate of $Δx^2$ in the Wasserstein distance ($W_1$-distance), whenever the initial data is decreasing and consists of a finite number of piecewise constants. It is shown that the Lax--Friedrichs, Enquist--Osher and Godunov schemes are $W_1$-contractive. Numerical experiments are presented to illustrate the main result. To the best of our knowledge, this is the first proof of second-order convergence of any numerical method for discontinuous solutions of nonlinear conservation laws.

Susanne Solem

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: $Δx^{1+1/p}$, $p \in [1,\infty]$.