  Traditionally, the Lighthill-Whitham-Richards (LWR) models for homogeneous
and inhomogeneous roads have been analyzed in flux-density space with the
fundamental diagram of the flux-density relation. In this paper, we present a
new framework for analyzing the LWR model, especially the Riemann problem at a
linear boundary in which the upstream and downstream links are homogeneous and
initially carry uniform traffic. We first review the definitions of local
supply and demand functions and then introduce the so-called supply-demand
diagram, on which a traffic state can be represented by its supply and demand,
rather than as density and flux as on a fundamental diagram. It is well-known
that the solutions to the Riemann problem at each link are self-similar with a
stationary state, and that the wave on the link is determined by the stationary
state and the initial state. In our new framework, there can also exist an
interior state next to the linear boundary on each link, which takes
infinitesimal space, and admissible conditions for the upstream and downstream
stationary and interior states can be derived in supply-demand space. With an
entropy condition consistent with a local supply-demand method in interior
states, we show that the stationary states exist and are unique within the
solution framework. We also develop a graphical scheme for solving the Riemann
problem, and the results are shown to be consistent with those in the
literature. We further discuss asymptotic stationary states on an inhomogeneous
ring road with arbitrary initial conditions and demonstrate the existence of
interior states with a numerical example. The framework developed in this study
is simpler than existing ones and can be extended for analyzing the traffic
dynamics in general road networks.

Analysis of PDEs