On structural properties of Helbing's gas-kinetic traffic flow model

There exist several types of models that describe the evolution of traffic flow on freeways and urban roads. In this paper we focus on some structural properties of one particular traffic flow model: the macroscopic, second-order gas-kinetic traffic flow model of Helbing. We will show that the 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 there are 1-shocks and 1-rarefaction waves, corresponding to the faster characteristic there are 2-shocks and 2-rarefaction waves. We also derive the formulas for the solution of the Riemann problem associated with this model in the equilibrium case, proving that the solution of this problem with density and flow non-negative in the initial condition on either side of the discontinuity cannot give rise to negative flow or density later on.