Axel Klar

Derivation of Continuum Traffic Flow Models from Microscopic Follow-the-Leader Models

In this paper we establish a connection between a microscopic follow-the-leader model based on ordinary differential equations and a semidiscretization of a macroscopic continuum model based on a conservation law. Naturally, it also turns out that the natural discretization of the conservation law in Lagrangian coordinates is equivalent to a straightforward time discretization of the microscopic model. We also show rigorously that, at least in the homogeneous case, the macroscopic model can be viewed as the limit of the time discretization of the microscopic model as the number of vehicles increases, with a scaling in space and time (a zoom) for which the density and the velocity remain fixed. Moreover, a numerical investigation and comparison is presented for the different models.

Gas flow in pipeline networks

We introduce a model for gas flow in pipeline networks based on the isothermal Euler equations. We model the intersection of multiple pipes by posing an additional assumption on the pressure at the interface. We give a method to obtain solutions to the gas network problem and present numerical results for sample networks.

A hierarchy of models for multilane vehicular traffic I: modeling

In the present paper multilane models for vehicular traffic are considered. A microscopic multilane model based on reaction thresholds is developed. Based on this model an Enskog- like kinetic model is developed. In particular, care is taken to incorporate the correlations between the vehicles. From the kinetic model a fluid dynamic model is derived. The macroscopic coefficients are deduced from the underlying kinetic model. Numerical simulations are presented for all three levels of description in [A. Klar and R. Wegener, SIAM J. Appl. Math., 59 (1999), pp. 1002--1011]. Moreover, a comparison of the results is given there.

Coupling conditions for gas networks governed by the isothermal Euler equations

We investigate coupling conditions for gas transport in networks where the governing equations are the isothermal Euler equations. We discuss intersections of pipes by considering solutions to Riemann problems. We introduce additional assumptions to obtain a solution near the intersection and we present numerical results for sample networks.

An Asymptotic-Induced Scheme for Nonstationary Transport Equations in the Diffusive Limit

An asymptotic-induced scheme for nonstationary transport equations with the diffusion scaling is developed. The scheme works uniformly for all ranges of mean-free paths. It is based on the asymptotic analysis of the diffusion limit of the transport equation. A theoretical investigation of the behavior of the scheme in the diffusion limit is given and an approximation property is proven. Moreover, numerical results for different physical situations are shown and the uniform convergence of the scheme is established numerically.

Modeling, Simulation, and Optimization of Traffic Flow Networks

A new model for highway traffic networks based on a detailed description of the junctions is presented. To obtain suitable conditions at the junctions, multilane equations are introduced and investigated. The new model is compared with currently known models for traffic flow networks for several situations. Finally, the model is used for network simulation and optimization.

Enskog-like kinetic models for vehicular traffic

In the present paper a general criticism of kinetic equations for vehicular traffic is given. The necessity of introducing an Enskog-type correction into these equations is shown. An Enskog-like kinetic traffic flow equation is presented and fluid dynamic equations are derived. This derivation yields new coefficients for the standard fluid dynamic equations of vehicular traffic. Numerical simulations for inhomogeneous traffic flow situations are shown together with a comparison between kinetic and fluid dynamic models.

CONGESTION ON MULTILANE HIGHWAYS

We present a new model for traffic on a multilane freeway (with n lanes). Our basic descriptors are the car density

Kinetic derivation of macroscopic anticipation models for vehicular traffic

\rho

SELF-PROPELLED INTERACTING PARTICLE SYSTEMS WITH ROOSTING FORCE

(in cars/mile), taken across all lanes in the freeway, and the average car velocity u (in miles/hour). The flux of cars across all lanes is given by