Godunov-like numerical fluxes for conservation laws on networks

TL;DR

This paper introduces two new Godunov-based numerical fluxes for discontinuous Galerkin schemes solving traffic flow models on networks, analyzing their properties and comparing with existing methods through numerical experiments.

Contribution

It presents novel Godunov-like fluxes at junctions for traffic flow models, ensuring conservation and realistic traffic distribution, with analysis and numerical validation.

Findings

Fluxes preserve conservation and traffic distribution properties.

In non-congested cases, flows follow driver preferences.

Small distribution errors occur in congested conditions, reflecting real-world factors.

Abstract

This paper deals with the construction of a discontinuous Galerkin scheme for the solution of Lighthill-Whitham-Richards traffic flows on networks. The focus of the paper is the construction of two new numerical fluxes at junctions, which are based on the Godunov numerical flux. We analyze the basic properties of the two Godunov-based fluxes and the resulting scheme, namely conservativity and the traffic distribution property. We prove that if the junction is not congested, the traffic flows according to predetermined preferences of the drivers. Otherwise a small traffic distribution error is present, which we interpret as either the existence of dedicated turning lanes, or factoring of human behavior into the model. We compare our approach to that of \v{C}ani\'c et al. (J. Sci. Comput., 2015). Numerical experiments are provided.