One-Dimensional Conservation Laws with Nonlocal Point Constraints on the Flux

Abstract

We review recent results and present new ones on one-dimensional conservation laws with point constraints on the flux. Their application is, for instance, the modeling of traffic flow through bottlenecks, such as exits in the context of pedestrians’ traffic and tollgates in vehicular traffic. In particular, we consider nonlocal constraints, which allow to model, e.g., the irrational behavior (“panic”) near the exits observed in dense crowds and the capacity drop at tollbooths in vehicular traffic. Numerical schemes for the considered applications, based on finite volume methods, are designed, their convergence is proved, and their validations are done with explicit solutions. Finally, we complement our results with numerical examples, which show that constrained models are able to reproduce important features in traffic flow, such as capacity drop and self-organization.