A nonlinear discrete velocity relaxation model for traffic flow
We derive a nonlinear 2-equation discrete velocity model for traffic flow from a continuous kinetic model. The model converges to scalar Lighthill--Whitham--Richards-type equations in the relaxation limit for all ranges of traffic data. Moreover, the model has an invariant domain appropriate for traffic flow modeling. It shows some similarities with the Aw--Rascle traffic model. However, the new model is simpler and yields, in the case of a concave fundamental diagram, an example of a totally linear degenerate hyperbolic relaxation model. We discuss the details of the hyperbolic main part and consider boundary conditions for the limit equations derived from the relaxation model. Stability of the model and the appearance of stable periodic solutions is investigated, as well as the cluster dynamics of the model for vanishing braking distance. Moreover, a relaxation scheme is developed based on the kinetic discrete velocity model. Finally, numerical results for various situations are presented, illustrating the analytical results.