Derivation of second order traffic flow models with time delays

Starting from microscopic follow-the-leader models, we develop hyperbolic delay partial differential equations to govern the density and velocity of vehicular traffic. The proposed models can be seen as an extension of the classical Aw-Rascle-Zhang model, where the reaction time of drivers appears as an additional term in the velocity equation. We propose numerical methods based on first principles and present a numerical study, where we focus on the impact of time delays in comparison to undelayed models.