Numerical simulations for the ARZ model for vehicular traffic with general point constraints on the density flux

In this paper we investigate numerically the model for vehicular traffic proposed in [B. Andreianov, C. Donadello, and M.D. Rosini, Math. Models Meth. Appl. Sci. 26(4):751–802, 2016]. The model is obtained by coupling the ARZ system with point constraints on the density flux, and reproduces traffic behavior at bottlenecks. We implement an adapted version of the Glimm scheme featuring a non-classical Riemann solver at the constraint locations. We show its numerical convergence and its ability to correctly capture the evolution of both conservative and Riemann invariant variables in all situations, including those involving the vacuum. Our scheme allows to conduct a number of numerical experiments to prove that the model reproduces several typical phenomena such as capacity drop, faster is slower effect and phantom jams. Beyond the theoretical results obtained in the aforementioned paper, we also consider point constraints whose values are determined by a Lipschitz continuous function of time, or depend non-locally on the solution itself.