Macroscopic traffic models: Shifting from densities to “Celerities”

• Published in: Applied mathematics and computation 2010-10, Vol.217 (3), p.963-971
• Main Author: Aubin, Jean-Pierre
• Format: Article
• Language: English
• Subjects: Basins; Calculus of variations and optimal control; Capture basins; Congestion; Daganzo variational principle; Density; Exact sciences and technology; Global analysis, analysis on manifolds; Global Positioning System; Lax–Hopf; Lighthill; Mathematical analysis; Mathematical models; Mathematics; Moskowitz; Newell; Numerical analysis; Numerical analysis. Scientific computation; Satellite navigation systems; Sciences and techniques of general use; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds; Traffic engineering; Traffic flow; Traffic function; Viability; Whitham and Richards equation
• ISSN: 0096-3003; 1873-5649
• DOI: 10.1016/j.amc.2010.02.032

The advent of techniques to measure velocities of “GPS equipped vehicles” using satellite technology, replacing the density of the road traffic sensors, motivates and justifies the revision of conceptual, mathematical algorithms and software based models. This paper summarizes studies on the traffic evolutions achieving the minimum of a congestion function controlled “macroscopic traffic velocities” called “celerities” instead of founding traffic regulation on the measures of traffic densities. The flux valued function is the Fenchel transform of the fundamental diagram and is a convex decreasing function. We use the properties of capture basins investigated in viability theory, specifically a Lax–Hopf formula characterizing them and the “Max-Plus” morphism of capture basins for deriving the statements proved in this paper. Traffic conditions involve as well boundary conditions or conditions on trajectories inside the domain.