## Section: New Results

### Macroscopic traffic flow models on networks

Participants : Guillaume Costeseque, Paola Goatin, Bhargava Rama Chilukuri [Georgia Tech, USA] , Maria Laura Delle Monache [U Rutgers - Camden] , Aurélien Duret [IFSTTAR, France] , Simone Göttlich [U Mannheim, Germany] , Oliver Kolb [U Mannheim, Germany] , Jorge A. Laval [Georgia Tech, USA] , Benedetto Piccoli [U Rutgers - Camden] , Armin Seyfried [Forschungszentrum Jülich, Germany] , Antoine Tordeux [Forschungszentrum Jülich, Germany] .

In collaboration with M.L. Delle Monache and B. Piccoli, and in the framework of the Associated Team ORESTE, we have introduced a new Riemann solver for traffic flow on networks. The Priority Riemann solver (PRS) provides a solution at junctions by taking into consideration priorities for the incoming roads and maximization of through flux. We prove existence of solutions for the solver for junctions with up to two incoming and two outgoing roads and show numerically the comparison with previous Riemann solvers. Additionally, we introduce a second version of the solver that considers the priorities as softer constraints and illustrate numerically the differences between the two solvers. See [24].

Still in collaboration with M.L. Delle Monache, we studied well-posedness of scalar conservation laws with moving flux constraints arising in the modeling of moving bottlenecks in traffic flow. In particular, we showed the Lipschitz continuous dependence of BV solutions with respect to the initial data and the constraint trajectory [23].

In collaboration with S. Göttlich and O. Kolb, we have investigated 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 have proposed 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 have also been addressed in a first discretize-then optimize framework [25].

Together with J. A. Laval and B. R. Chilukuri, we have investigated the implications of source terms in the Hamilton-Jacobi formulation of macroscopic first order traffic flow models. Hamilton-Jacobi equations (without source terms) have been demonstrated to be very useful in traffic flow engineering since they provide explicit formula for initial and boundary-values problems. However, for sake of realism, additional source terms should be incorporated to account for continuous inflows or outflows on freeways for instance. We showed that explicit Lax-Hopf formula can still be obtained when the source term is exogenous, say the lateral inflow or outflow does not depend on the density on the main road. We also provide numerical methods based on Bellman's dynamic programming principle to deal with non-exogenous source terms in discrete time [7].

With A. Duret, we have designed a new traffic flow model for taking into account the multiclass and multilane features of real traffic. This model is based on a system of coupled Hamilton-Jacobi PDEs for an appropriate choice of framework that mixes spatial and Lagrangian coordinates. The coupling conditions emerge from the moving bottleneck theory that has been developed in the traffic flow literature several years ago but for which a real mathematical sound basis lacked. Very recently, there were some new results dealing with the existence of a solution under suitable assumptions [64]. However, these results were set for the hyperbolic conservation law in Eulerian coordinates and they are not straightforward to be extended to Hamilton-Jacobi equations in different coordinates. Despite that the well-posedness of the problem is still an open problem, a numerical method is developed by taking advantage of the classical representation formula available for HJ PDEs. This numerical scheme has been proved to provide good qualitative results [14].

In collaboration with A. Tordeux, M. Herty and A. Seyfried, we studied the derivation of convection-diffusion macroscopic traffic flow models from a first order microscopic follow-the-leader model that takes into account a non-trivial time delay. The derivation is based on a change of variables from Lagrangian to Eulerian coordinates and makes use of Taylor expansions with respect to the time delay. The macroscopic diffusion term is due to the microscopic reaction time parameter and allows to reproduce the scatter of empirical flow-density data. Different numerical methods are proposed for computing the numerical flux and the linear stability of the homogeneous solutions obtained for each method is investigated. Interestingly, we recover some stability results for infinite systems of delayed ODEs [27].