Continuous kinematic wave models of merging traffic flow
  Merging junctions are important network bottlenecks, and a better understanding of merging traffic dynamics has both theoretical and practical implications. In this paper, we present continuous kinematic wave models of merging traffic flow which are consistent with discrete Cell Transmission Models with various distribution schemes. In particular, we develop a systematic approach to constructing kinematic wave solutions to the Riemann problem of merging traffic flow in supply-demand space. In the new framework, Riemann solutions on a link consist of an interior state and a stationary state, subject to admissible conditions such that there are no positive and negative kinematic waves on the upstream and downstream links respectively. In addition, various distribution schemes in Cell Transmission Models are considered entropy conditions. In the proposed analytical framework, we prove that the stationary states and boundary fluxes exist and are unique for the Riemann problem for both fair and constant distribution schemes. We also discuss two types of invariant merge models, in which local and discrete boundary fluxes are the same as global and continuous ones. With numerical examples, we demonstrate the validity of the analytical solutions of interior states, stationary states, and corresponding kinematic waves. Discussions and future studies are presented in the conclusion section. 
Wen-Long Jin