This paper deals with the Aw-Rascle-Zhang [2,31] model for traffic flow on unidirectional road networks. We construct weak solutions to Riemann problems at the junctions, which conserve the mass and the generalized momentum. In particular, we introduce a new approach to approximate the homogenized pressure through an additional equation for the propagation of a reference pressure. The resulting system of coupled conservation laws is then solved using an appropriate numerical scheme of Godunov type. Numerical simulations show that the proposed approach enables to approximate the homogenized pressure sufficiently well. The features of the new approach are illustrated through a comparative analysis with other methods proposed in the literature, namely [13,19,16,23,28], for the second order traffic model [2,31] and the Lighthill-Whitham-Richards model [26,27].

The transition from a microscopic model for the movement of many particles to a macroscopic continuum model for a density flow is studied. The microscopic model for the free flow is completely deterministic, described by an interaction potential that leads to a coherent motion where all particles move in the same direction with the same speed known as a flock. Interaction of the flock with boundaries, obstacles and other flocks leads to a temporary destruction of the coherent motion that macroscopically can be modeled through density dependent diffusion. The resulting macroscopic model is an advection-diffusion equation for the particle density whose diffusion coefficient is density dependent. Examples describing i) the interaction of material flow on a conveyor belt with an obstacle that redirects or restricts the material flow and ii) the interaction of flocks (of fish or birds) with boundaries and iii) the scattering of two flocks as they bounce off each other are discussed. In each case, the advection-diffusion equation is strictly hyperbolic before and after the interaction while the interaction phase is described by a parabolic equation. A numerical algorithm to solve the advection-diffusion equation through the transition is presented.

The model under consideration is a non‐local conservation law in two space dimensions that might be used to model material or pedestrian flow. We accelerate the simulation of the non‐local model using parallelization of the non‐local terms in the programming language Haskell. The implementation allows for the computation of the solution to the conservation law with a finite volume method at reasonable computational cost even for large domains and fine discretizations.

We propose a space mapping-based optimization algorithm for microscopic interacting particle dynamics which are inappropriate for direct optimization. This is of relevance for example in applications with bounded domains such that the microscopic optimization is difficult. The space mapping algorithm exploits the relationship of the microscopic description of the interacting particle system and the corresponding macroscopic description as partial differential equation in the "many particle limit". We validate the approach with the help of a toy problem that allows for direct optimization. Then we study the performance of the algorithm in two applications. An evacuation dynamic is considered and the transportation of goods on a conveyor belt is optimized. The numerical results underline the feasibility of the proposed algorithm.

Second order macroscopic traffic flow models are able to reproduce the so-called capacity drop effect, i.e., the phenomenon that the outflow of a congested region is substantially lower than the maximum achievable flow. Within this work, we propose a first order model for a junction with ramp buffer that is solely modified at the intersection so that the capacity drop is captured. Theoretical investigations motivate the new choice of coupling conditions and illustrate the difference to purely first and second order models. The numerical example considering the optimal control of the onramp merging into a main road highlights that the combined model generates similar results as the second order model.