  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.
