Networks of hyperbolic balance laws

Models with networks of hyperbolic balance laws are currently used for several applications, e.g. river flows, the human circulatory system, gas pipelines or road networks. These motivations triggered a constant development of analytical results as well as numerical methods. This talk aims to provide an overview of the developments in the past years with a focus on the construction of suitable numerical schemes. Additionally a possible extension to non-conservative equations will discussed.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

Kinetic layers and coupling conditions for macroscopic equations on networks I: the wave equation

We consider kinetic and associated macroscopic equations on networks. The general approach will be explained in this paper for a linear kinetic BGK model and the corresponding limit for small Knudsen number, which is the wave equation. Coupling conditions for the macroscopic equations are derived from the kinetic conditions via an asymptotic analysis near the nodes of the network. This analysis leads to the consideration of a fixpoint problem involving the coupled solutions of kinetic half-space problems. A new approximate method for the solution of kinetic half-space problems is derived and used for the determination of the coupling conditions. Numerical comparisons between the solutions of the macroscopic equation with different coupling conditions and the kinetic solution are presented for the case of tripod and more complicated networks

Kinetic layers and coupling conditions for nonlinear scalar equations on networks

We consider a kinetic relaxation model and an associated macroscopic scalar nonlinear hyperbolic equation on a network. Coupling conditions for the macroscopic equations are derived from the kinetic coupling conditions via an asymptotic analysis near the nodes of the network. This analysis leads to the combination of kinetic half-space problems with Riemann problems at the junction. Detailed numerical comparisons between the different models show the agreement of the coupling conditions for the case of tripod networks

A nonlinear discrete-velocity relaxation model for traffic flow

We derive a nonlinear 2-equation discrete-velocity model for traffic flow from a continuous kinetic model. The model converges to scalar Lighthill-Whitham type equations in the relaxation limit for all ranges of traffic data. Moreover, the model has an invariant domain appropriate for traffic flow modeling. It shows some similarities with the Aw-Rascle traffic model. However, the new model is simpler and yields, in case of a concave fundamental diagram, an example for a totally linear degenerate hyperbolic relaxation model. We discuss the details of the hyperbolic main part and consider boundary conditions for the limit equations derived from the relaxation model. Moreover, we investigate the cluster dynamics of the model for vanishing braking distance and consider a relaxation scheme build on the kinetic discrete velocity model. Finally, numerical results for various situations are presented, illustrating the analytical results

Kinetic derivation of a Hamilton-Jacobi traffic flow model

Kinetic models for vehicular traffic are reviewed and considered from the point of view of deriving macroscopic equations. A derivation of the associated macroscopic traffic flow equations leads to different types of equations: in certain situations modified Aw-Rascle equations are obtained. On the other hand, for several choices of kinetic parameters new Hamilton-Jacobi type traffic equations are found. Associated microscopic models are discussed and numerical experiments are presented discussing several situations for highway traffic and comparing the different models

A class of multi-phase traffic theories for microscopic, kinetic and continuum traffic models

In the present paper a review and numerical comparison of a special class of multi-phase traffic theories based on microscopic, kinetic and macroscopic traffic models is given. Macroscopic traffic equations with multi-valued fundamental diagrams are derived from different microscopic and kinetic models. Numerical experiments show similarities and differences of the models, in particular, for the appearance and structure of stop and go waves for highway traffic in dense situations. For all models, but one, phase transitions can appear near bottlenecks depending on the local density and velocity of the flow

Numerical schemes for networks of hyperbolic conservation laws

In this paper we propose a procedure to extend classical numerical schemes for hyperbolic conservation laws to networks of hyperbolic conservation laws. At the junctions of the network we solve the given coupling conditions and minimize the contributions of the outgoing numerical waves. This flexible procedure allows us to also use central schemes at the junctions. Several numerical examples are considered to investigate the performance of this new approach compared to the common Godunov solver and exact solutions