An easy-to-use algorithm for simulating traffic flow on networks: Theoretical study

Abstract

Abstract. In this paper we study a model for traffic flow on networks based on a hyperbolic system of conservation laws with discontinuous flux. Each equation describes the density evolution of vehicles having a common path along the network. In this formulation the junctions disappear since each path is considered as a single uninterrupted road.We consider a Godunov-based approximation scheme for the system which is very easy to implement. Besides basic properties like the conservation of cars and positive bounded solutions, the scheme exhibits other nice properties, being able to select automatically a solution at junctions without requiring external procedures (e.g., maximization of the flux via a linear programming method). Moreover, the scheme can be interpreted as a discretization of the traffic models with buffer, although no buffer is introduced here.Finally, we show how the scheme can be recast in the framework of the classical theory of traffic flow on networks, where a conservation law has to be solved on each arc of the network. This is achieved by solving the Riemann problem for a modified equation, and showing that its solution corresponds to the one computed by the numerical scheme.

Transport of measures on networks

“…This approach differs from the one proposed in [8], which instead assigns a path to each microscopic vehicle through the network in the spirit of the multipath traffic model introduced in [3,4]. 6.2.…”

An interface-free multi-scale multi-order model for traffic flow

“…In this context let us mention the paper by Cristiani and Sahu [14] which investigates the many-particle limit of a first-order FtL model suitably extended to a road network. The corresponding macroscopic model appears to be the extension of the LWR model on networks introduced by Hilliges and Weidlich [32] and then extensively studied by Bretti et al [4] and Briani and Cristiani [6].…”

Sensitivity analysis of the LWR model for traffic forecast on large networks using Wasserstein distance

“…In general, they all allow to determine a unique solution for the traffic evolution on the network, but the solutions might be different. In the following we will employ the approach (iii) together with a Godunov-based numerical approximation [4,5], also used to reproduce Wardrop equilibria [8]. More recently, it was shown that the solution computed by the path-based model coincides with the many-particle limit of the first-order microscopic follow-the-leader model [9].…”

“…In Sec. 2 we present in detail the mathematical model, which is a local version of the path-based approach [4,5], suitable to deal with large networks. In Sec.…”

“…Clearly 0 ≤ α v rr ≤ 1 ∀r,r and r α v rr = 1 ∀r. Now, following the path-based approach [4,5], we look at all possible paths across the vertex v. Having n v inc incoming edges and n v out outgoing edges, we have n v p := n v inc × n v out possible paths. We denote by µ n e,j (p,v) the density of the vehicles in the cell j of edge e at time t n moving along path p across vertex v.…”