Analytical and numerical investigations of refined macroscopic traffic flow models
Michael Herty Reinhard Illner
Kinetic & Related Models 2010, 3(2): 311-333 doi: 10.3934/krm.2010.3.311
We continue research on generalized macroscopic models of conservation type as started in [15]. In this paper we keep the characteristic (for traffic) non-locality removed in [15] by Taylor expansion and discuss the merits and problems of such an expansion. We observe that the models satisfy maximum principles and conclude that "triggers'' are needed in order to cause traffic jams (braking waves) in traffic guided by such models. Several such triggers are introduced and discussed. The models are refined further in order to properly address non-monotonic (in speed) traffic regimes, and the inclusion of an individual reaction time is discussed in the context of a braking wave. A number of numerical experiments are conducted to exhibit our findings.
keywords: Fokker-Planck Models Mathematical Modeling. Traffic Flow
On Stop-and-Go waves in dense traffic
Michael Herty Reinhard Illner
Kinetic & Related Models 2008, 1(3): 437-452 doi: 10.3934/krm.2008.1.437
From a Vlasov-type kinetic equation with nonlocal braking and acceleration forces, taken as a traffic model for higher densities, we derive macroscopic equations generalizing the second order model of conservation laws suggested by Aw and Rascle [1] and Zhang [19]. The nonlocality remains present in these equations, but more conventional, local equations are derived by using suitable Taylor expansion. A second order model of this type is discussed in some detail and is shown to possess traveling wave solutions that resemble stop-and-go waves in dense traffic. A phase space analysis suggests that inside the class of such traveling waves there are steady solutions that are stable.
keywords: nonlinear stability stop-and-go waves traffic flow nonlocal equations.
Coupling of non-local driving behaviour with fundamental diagrams
Michael Herty Reinhard Illner
Kinetic & Related Models 2012, 5(4): 843-855 doi: 10.3934/krm.2012.5.843
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.
keywords: Traffic flow mathematical modeling.
Analysis and simulations of a refined flocking and swarming model of Cucker-Smale type
Martial Agueh Reinhard Illner Ashlin Richardson
Kinetic & Related Models 2011, 4(1): 1-16 doi: 10.3934/krm.2011.4.1
The Cucker-Smale model for flocking or swarming of birds or insects is generalized to scenarios where a typical bird will be subject to a) a friction force term driving it to fly at optimal speed, b) a repulsive short range force to avoid collisions, c) an attractive "flocking" force computed from the birds seen by each bird inside its vision cone, and d) a "boundary" force which will entice birds to search for and return to the flock if they find themselves at some distance from the flock. We introduce these forces in detail, discuss the required cutoffs and their implications and show that there are natural bounds in velocity space. Well-posedness of the initial value problem is discussed in spaces of measure-valued functions. We conclude with a series of numerical simulations.
keywords: swarming particle model kinetic equation. Flocking
Remarks on a class of kinetic models of granular media: Asymptotics and entropy bounds
Martial Agueh Guillaume Carlier Reinhard Illner
Kinetic & Related Models 2015, 8(2): 201-214 doi: 10.3934/krm.2015.8.201
We obtain new a priori estimates for spatially inhomogeneous solutions of a kinetic equation for granular media, as first proposed in [3] and, more recently, studied in [1]. In particular, we show that a family of convex functionals on the phase space is non-increasing along the flow of such equations, and we deduce consequences on the asymptotic behaviour of solutions. Furthermore, using an additional assumption on the interaction kernel and a ``potential for interaction'', we prove a global entropy estimate in the one-dimensional case.
keywords: global in time estimates Kinetic granular media asymptotic behavior entropy bounds.

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