"A note on the Entropy Solutions of the Hydrodynamic Model of Traffic Flow" revisited

Raphaele Herbin (1) , Ludovic Leclercq (2)

1 LATP - Laboratoire d'Analyse, Topologie, Probabilités
2 LICIT - Laboratoire d'ingénierie circulation transports

A central question in [Velan and Florian(2002)] is the influence of a non differentiable fundamental diagram on the solutions of the LWR model. This question is crucial because experimental observations put credit on piecewise linear fundamental diagrams (PLFD) [Leclercq(2005), Chiabaut et al.(2009)] and especially on triangular ones. In [Velan and Florian(2002)] it is claimed that, with the latter diagrams, the solution of the LWR model is unique but non-entropic. This note aims to invalidate this result. Considering a triangular fundamental diagram, we will demonstrate (i) that the weak solutions of the LWR are not unique and (ii) that the solution which is usually retained (and claimed to be unique) is in fact the unique weak entropy solution in the sense of Kruskov [Kruˇzkov(1970)]. This means that PLFD cannot be disproved on an alleged mathematical basis, since contrary to what is claimed in [Velan and Florian(2002)], the entropy criterion is indeed respected by its solutions.

Domaines: Analyse numérique [math.NA]

Raphaele Herbin, Ludovic Leclercq. "A note on the Entropy Solutions of the Hydrodynamic Model of Traffic Flow" revisited. Transportation Science, 2011, 45 (1), pp.138-142. ⟨hal-00793622⟩