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arXiv · 1602.00459

Second-order convergence of monotone schemes for conservation laws

Ulrik S. Fjordholm·Susanne Solem
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Abstract

We prove that a class of monotone, \emph{$W_1$-contractive} schemes for scalar conservation laws converge at a rate of $\Delta x^2$ 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.

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Ulrik S. Fjordholm, Susanne Solem. 2016-02-01. Second-order convergence of monotone schemes for conservation laws. https://doi.org/10.1137/16m1059138

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