Macroscopic Models with Dynamic Velocity
In the macroscopic first-order (Lighthill-Whitham-Richards, LWR) models presented in the previous two chapters, the local speed and flow are statically coupled to the density by the fundamental relation. This implies instantaneous adaptation to new circumstances and leads to unbounded accelerations and other unrealistic consequences such as the lack of hysteresis effects or traffic instabilities. In the second-order models considered in this chapter, the local speed possesses its own dynamical acceleration equation describing speed changes as a function of density, local speed, their gradients, and possibly other exogenous factors. Second-order models are the models of choice to macroscopically describe traffic-flow instabilities leading to traffic waves, the capacity drop phenomenon, or scattered flow-density data. Besides discussing representative models, this chapter describes approximative numerical integration techniques which are more demanding than that of the LWR models.