A class of germs arising from homogenization in traffic flow on junctions

TL;DR

This paper rigorously derives effective junction conditions for traffic flow models at large scales by homogenizing mesoscopic traffic laws with periodic controls, identifying a class of germs that characterize the macroscopic behavior.

Contribution

It introduces a novel homogenization approach for traffic flow junctions, identifying a general class of effective junction conditions called germs, and constructs explicit correctors for these conditions.

Findings

Homogenization of traffic flow junctions yields a class of effective conditions.

Identification of a characteristic subgerm that determines the overall germ.

Explicit construction of correctors related to the subgerm using Hamilton-Jacobi equations.

Abstract

We consider traffic flows described by conservation laws. We study a 2:1 junction (with two incoming roads and one outgoing road) or a 1:2 junction (with one incoming road and two outgoing roads). At the mesoscopic level, the priority law at the junction is given by traffic lights, which are periodic in time and the traffic can also be slowed down by periodic in time flux-limiters. After a long time, and at large scale in space, we intuitively expect an effective junction condition to emerge. Precisely, we perform a rescaling in space and time, to pass from the mesoscopic scale to the macroscopic one. At the limit of the rescaling, we show rigorous homogenization of the problem and identify the effective junction condition, which belongs to a general class of germs (in the terminology of [6, 21, 37]). The identification of this germ and of a characteristic subgerm which determines the whole germ, is the first key result of the paper. The second key result of the paper is the construction of a family of correctors whose values at infinity are related to each element of the characteristic subgerm. This construction is indeed explicit at the level of some mixed Hamilton-Jacobi equations for concave Hamiltonians (i.e. fluxes). The explicit solutions are found in the spirit of representation formulas for optimal control problems.