Growth and decay of shock and acceleration waves in a traffic flow model with relaxation

The usual Ficken-based constitutive relation for traffic flux is replaced with one based on the Maxwell–Cattaneo model. The resulting flux law, which now takes into account the reaction time of driver and vehicle, results in a second-order, hyperbolic generalization of Burgers’ equation as the PDE governing the traffic density. An analytical study of this equation is presented with an emphasis on shock and related kinematic wave phenomena. Specifically, the exact traveling wave solution (TWS) is derived and it is shown that shock formation is possible only if the diffusivity is non-vanishing. Using singular surface theory, exact amplitude expressions for both shock and acceleration waves are obtained and their temporal evolution determined. The exact upper bound of the reaction time parameter is established and connections between singular surface and TWS results are noted.