A Fast Lax-Hopf formula to solve the Lighthill-Whitham-Richards traffic flow model on networks

TL;DR

This paper introduces a fast, semi-analytical Lax-Hopf algorithm for solving the LWR traffic flow model on networks, reducing computational complexity while maintaining accuracy for various fundamental diagrams.

Contribution

The paper extends the Lax-Hopf algorithm to efficiently compute boundary flows in traffic networks, enabling faster simulations for a wide range of fundamental diagrams and initial conditions.

Findings

The proposed algorithm reduces computational complexity compared to existing methods.

It performs comparably to the Link Transmission Model in traffic network simulations.

The method is applicable to general concave and triangular fundamental diagrams.

Abstract

Efficient and exact algorithms are important for performing fast and accurate traffic network simulations with macroscopic traffic models. In this paper, we extend the semi-analytical Lax-Hopf algorithm in order to compute link inflows and outflows with the LWR model. Our proposed Fax Lax-Hopf algorithm has a very low computational complexity. We demonstrate that some of the original algorithm's operations (associated with the initial conditions) can be discarded, leading to a faster computation of boundary demand/supplies in network simulation problems, for general concave fundamental diagrams. Moreover, the computational cost can be further reduced for triangular Fundamental Diagrams and specific space-time discretizations. The resulting formulation has a performance comparable to the Link Transmission Model and, since it solves the original LWR model for a wide range of FD shapes, with any initial configuration, it is suitable to solve a broad range of traffic operations problems. As part of the analysis, we compare the performance of the proposed scheme to other well-known computational methods.