Discrete-velocity models and relaxation schemes for traffic flows

We present simple discrete-velocity models for traffic flows. The novel feature in the corresponding relaxation system is the presence of nonnegative velocities only. We show that in the small relaxation limit the discrete models reduce to the Lighthill-Whitham-Richards equation. In addition we propose second order schemes combined with implicit-explicit time integrators as proper discretization of the relaxation-type system. Numerical tests are carried out on various situations in traffic flow. The results show that the proposed models are able to describe correctly the formation of backward waves induced by traffic jam.