Continuous kinematic wave models of merging traffic flow

Traffic dynamics at a merging junction can be numerically solved with discrete conservation equationsand so-called supply-demand methods. In this paper, we first introduce a continuous multi-commoditykinematic wave model of merging traffic and then develop a new framework for constructing the solutionsto its Riemann problem with jump initial conditions. In the supply-demand space, the solutions on a linkconsist of an interior state and a stationary state, subject to admissible conditions such that there are nopositive and negative kinematic waves on the upstream and downstream links respectively. In addition, thesolutions have to satisfy entropy conditions defined by the supply-demand method in the interior states anda corresponding distribution scheme. For a merging junction with two upstream links, we prove that thestationary states and boundary fluxes exist and are unique for the Riemann problem for both fair and con-stant distribution schemes. With a numerical example, we demonstrate that the boundary fluxes convergeto the analytical solutions at any positive time when we decrease the period of a time interval.