A distributed model of traffic flows on extended regions

This work deals with the modelling of traffic flows in complex networks, spanning two-dimensional regions whose size ( macroscale ) is much greater than the characteristic size of the network arcs ( microscale). A typical example is the modelling of traffic flow in large urbanized areas with diameter of hundreds of kilometers, where standard models of traffic flows on networks resolving all the streets are computationally too expensive. Starting from a stochastic lattice gas model with simple constitutive laws, we derive a distributed two-dimensional model of traffic flow, in the form of a non-linear diffusion-advection equation for the particle density. The equation is formally equivalent to a (non-linear) Darcy's filtration law. In particular, it contains two parameters that can be seen as the porosity and the permeability tensor of the network. We provide suitable algorithms to extract these parameters starting from the geometry of the network and a given microscale model of traffic flow (for instance based on cellular automata). Finally, we compare the fully microscopic simulation with the finite element solution of our upscaled model in realistic cases, showing that our model is able to capture the large-scale feature of the flow.