  This paper addresses an open problem in traffic modeling: the second-order
macroscopic node problem. A second-order macroscopic traffic model, in contrast
to a first-order model, allows for variation of driving behavior across
subpopulations of vehicles in the flow. The second-order models are thus more
descriptive (e.g., they have been used to model variable mixtures of
behaviorally-different traffic, like car/truck traffic, autonomous/human-driven
traffic, etc.), but are much more complex. The second-order node problem is a
particularly complex problem, as it requires the resolution of discontinuities
in traffic density and mixture characteristics, and solving of throughflows for
arbitrary numbers of input and output roads to a node (in other words, this is
an arbitrary-dimensional Riemann problem with two conserved quantities). In
this paper, we extend the well-known "Generic Class of Node Model" constraints
to the second order and present a simple solution algorithm to the second-order
node problem. Our solution makes use of a recently-introduced dynamic system
characterization of the first-order node model problem, which gives insight and
intuition as to the continuous-time dynamics implicit in node models. We
further argue that the common "supply and demand" construction of node models
that decouples them from link models is not suitable to the second-order node
problem. Our second-order node model and solution method have immediate
applications in allowing modeling of behaviorally-complex traffic flows of
contemporary interest (like partially-autonomous-vehicle flows) in arbitrary
road networks.
