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.
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