Riemann solver for a kinematic wave traffic model with discontinuous flux

TL;DR

This paper develops a Riemann solver for a traffic flow model with discontinuous flux, using a mollifier and a Godunov-type scheme to accurately handle zero waves and improve numerical stability.

Contribution

It introduces a novel Riemann solver for discontinuous flux models and a Godunov-type scheme that effectively manages zero waves with infinite speed.

Findings

The proposed method accurately captures zero waves and their effects.

Numerical results show convergence and improved stability.

Comparison with WENO scheme validates effectiveness.

Abstract

We investigate a model for traffic flow based on the Lighthill-Whitham-Richards model that consists of a hyperbolic conservation law with a discontinuous, piecewise-linear flux. A mollifier is used to smooth out the discontinuity in the flux function over a small distance epsilon << 1 and then the analytical solution to the corresponding Riemann problem is derived in the limit as epsilon goes to 0. For certain initial data, the Riemann problem can give rise to zero waves that propagate with infinite speed but have zero strength. We propose a Godunov-type numerical scheme that avoids the otherwise severely restrictive CFL constraint that would arise from waves with infinite speed by exchanging information between local Riemann problems and thereby incorporating the effects of zero waves directly into the Riemann solver. Numerical simulations are provided to illustrate the behaviour of zero waves and their impact on the solution. The effectiveness of our approach is demonstrated through a careful convergence study and comparisons to computations using a third-order WENO scheme.