Coupling of Non-local Driving Behaviour with Fundamental Diagrams

We present an extended discussion of a macroscopic traffic flow model [18] which includes non-local and relaxation terms for vehicular traffic flow on unidirectional roads. The braking and acceleration forces are based on a behavioural model and on free flow dynamics. The latter are modelled by using different fundamental diagrams. Numerical investigations for different situations illustrate the properties of the mathematical model. In particular, the emergence of stop-and-go waves is observed for suitable parameter ranges. 1. Motivation and Introduction. We are concerned with refinements of the non-local traffic models previously introduced in [17] and [18], and this paper is a continuation of [18]. The studies done there show how non-local macroscopic models can be derived from kinetic models of Vlasov-type. They further contain a number of analytical and numerical investigations concerning the qualitative behaviour of solutions, such as the formation and propagation of traveling wave solutions, the impact of a local speed limit, and the origin of stop-and-go waves. In [22] the functional-differential equation arising for the propagation of a traveling braking wave is discussed as an interesting mathematical object in its own right. The models are generalizations of the macroscopic model first suggested by AwRascle [1] and Zhang [39]. In the form considered in [18] we observed that maximum principles are satisfied; while this is a pleasant feature from a mathematical point of view, it is rather unrealistic in real traffic. For example, consider a situation where traffic moves at a constant large speed u and a very high density ρ. The constant pair (ρ, u) will formally be a steady solution of the models from [18], but in reality drivers in such a situation would (because of safety concerns) make an effort to reduce density and speed and would slow down to this end. In versions of the Aw-Rascle model this is usually achieved by the inclusion of a relaxation term based on a fundamental diagram. In the simplest form, this means that in the absence of other stimuli drivers will brake or accelerate towards a comfortable 2000 Mathematics Subject Classification. Primary: 90B20; Secondary: 35L65.