I. Necoara, B. De Schutter, and J. Hellendoorn, "Structural properties of Helbing's traffic flow model," Transportation Research Record, no. 1883, pp. 21-30, 2004.
This paper analyzes the structural properties of the shock and rarefaction wave solutions of a macroscopic, second-order non-local continuum traffic flow model, namely Helbing's model. We will show that this model has two families of characteristics for the shock wave solutions: one characteristic is slower, and the other one is faster than the average vehicle speed. Corresponding to the slower characteristic we have 1-shocks and 1-rarefaction waves, the behavior of which is similar to that of shocks and rarefaction waves in the first-order model of Lighthill-Whitham-Richards. Corresponding to the faster characteristic there are 2-shocks and 2-rarefaction waves, which behave differently from the previous one, in the sense that the information in principle travels faster than average vehicle speed, but - as we shall see - in Helbing's model this inconsistency is solved via the addition of a non-local term. We will show that for the Helbing model the shocks do not produce negative states as other second-order models do. In this paper we also derive the formulas for the solution of the Riemann problem associated with this model in the equilibrium case.