A Riemann solver for a system of hyperbolic conservation laws at a general road junction

TL;DR

This paper introduces a novel Riemann solver for hyperbolic conservation laws at road junctions, enhancing traffic flow modeling by ensuring unique solutions and consistent flux functions using interior states and a local flux approach.

Contribution

It proposes a new Riemann solver that incorporates interior states and a local flux function, improving the modeling of traffic flow at complex junctions.

Findings

Unique solution for the Riemann problem at junctions.

The flux function is invariant and Godunov.

The method ensures fair merging and FIFO diverging rules.

Abstract

The kinematic wave model of traffic flow on a road network is a system of hyperbolic conservation laws, for which the Riemann solver is of physical, analytical, and numerical importance. In this paper, we present a Riemann solver at a general network junction. In the Riemann solver, we replace the entropy condition in [25] by a local, discrete flux function used in Cell Transmission Model [11]. To enable such an entropy condition, which is consistent with fair merging and first-in-first-out diverging rules, we enlarge the weak solution space by introducing interior states on a set of measure zero, associated with stationary discontinuities at the junction. In the demand-supply space, we demonstrate that the Riemann problem is uniquely solved, in the sense that stationary states and, therefore, kinematic waves on all links can be uniquely determined from feasible conditions on both stationary and interior states as well as the entropy condition that prescribes boundary fluxes from interior states. In addition, the resulting global flux function is the same as the local one. Thus the flux function is both invariant and Godunov.