Two algorithms for a fully coupled and consistently macroscopic PDE-ODEsystem modeling a moving bottleneck on a road

In this paper we propose two numerical algorithms to solve a coupled PDE-ODE systemwhich models a slow vehicle (bottleneck) moving on a road together with other cars. The resultingsystem is fully coupled because the dynamics of the slow vehicle depends on the density of cars and,at the same time, it causes a capacity drop in the road, thus limiting the car flux. The first algorithm,based on the Wave Front Tracking method, is suitable for theoretical investigations and convergenceresults. The second one, based on the Godunov scheme, is used for numerical simulations. The caseof multiple bottlenecks is also investigated.