  In this paper a novel non-negative finite volume discretization scheme is
proposed for certain first order nonlinear partial differential equations
describing conservation laws arising in traffic flow modelling. The spatially
discretized model is shown to preserve several fundamentally important
analytical properties of the conservation law (e.g., conservativeness,
capacity) giving rise to a set of (second order) polynomial ODEs. Furthermore,
it is shown that the discretized traffic flow model is formally kinetic and
that it can be interpreted in a compartmental context. As a consequence,
traffic networks can be represented as reaction graphs. It is shown that the
model can be equipped with on- and off- ramps in a physically meaningful way,
still preserving the advantageous properties of the discretization. Numerical
case studies include empirical convergence tests, and the stability analysis
presented in the paper paves the way to scalable observer and controller
design.
