The link transmission model (LTM) has great potential for simulating traffic
flow in large-scale networks since it is much more efficient and accurate than
the Cell Transmission Model (CTM). However, there lack general continuous
formulations of LTM, and there has been no systematic study on its analytical
properties such as stationary states and stability of network traffic flow. In
this study we attempt to fill the gaps. First we apply the Hopf-Lax formula to
derive Newell's simplified kinematic wave model with given boundary cumulative
flows and the triangular fundamental diagram. We then apply the Hopf-Lax
formula to define link demand and supply functions, as well as link queue and
vacancy functions, and present two continuous formulations of LTM, by
incorporating boundary demands and supplies as well as invariant macroscopic
junction models. With continuous LTM, we define and solve the stationary states
in a road network. We also apply LTM to directly derive a Poincar\'e map to
analyze the stability of stationary states in a diverge-merge network. Finally
we present an example to show that LTM is not well-defined with non-invariant
junction models. We can see that Newell's model and LTM complement each other
and provide an alternative formulation of the network kinematic wave model.
This study paves the way for further extensions, analyses, and applications of
LTM in the future.Comment: 27 pages, 5 figure