NHM
Capacity drop and traffic control for a second order traffic model
Oliver Kolb Simone Göttlich Paola Goatin
Networks & Heterogeneous Media 2017, 12(4): 663-681 doi: 10.3934/nhm.2017027

In this paper, we illustrate how second order traffic flow models, in our case the Aw-Rascle equations, can be used to reproduce empirical observations such as the capacity drop at merges and solve related optimal control problems. To this aim, we propose a model for on-ramp junctions and derive suitable coupling conditions. These are associated to the first order Godunov scheme to numerically study the well-known capacity drop effect, where the outflow of the system is significantly below the expected maximum. Control issues such as speed and ramp meter control are also addressed in a first-discretize-then-optimize framework.

keywords: Traffic flow second order model on-ramp coupling numerical simulations optimal control
NHM
Optimization for a special class of traffic flow models: Combinatorial and continuous approaches
Simone Göttlich Oliver Kolb Sebastian Kühn
Networks & Heterogeneous Media 2014, 9(2): 315-334 doi: 10.3934/nhm.2014.9.315
In this article, we discuss the optimization of a linearized traffic flow network model based on conservation laws. We present two solution approaches. One relies on the classical Lagrangian formalism (or adjoint calculus), whereas another one uses a discrete mixed-integer framework. We show how both approaches are related to each other. Numerical experiments are accompanied to show the quality of solutions.
keywords: Traffic networks conservation laws combinatorial optimization. control of discretized PDEs adjoint calculus
NHM
A Godunov type scheme for a class of LWR traffic flow models with non-local flux
Jan Friedrich Oliver Kolb Simone Göttlich
Networks & Heterogeneous Media 2018, 13(4): 531-547 doi: 10.3934/nhm.2018024

We present a Godunov type numerical scheme for a class of scalar conservation laws with non-local flux arising for example in traffic flow models. The proposed scheme delivers more accurate solutions than the widely used Lax-Friedrichs type scheme. In contrast to other approaches, we consider a non-local mean velocity instead of a mean density and provide $L^∞$ and bounded variation estimates for the sequence of approximate solutions. Together with a discrete entropy inequality, we also show the well-posedness of the considered class of scalar conservation laws. The better accuracy of the Godunov type scheme in comparison to Lax-Friedrichs is proved by a variety of numerical examples.

keywords: Scalar conservation laws non-local flux Godunov scheme traffic flow models numerical simulations

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