Godunov-like numerical fluxes for conservation laws on networks

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.