Concentration in vanishing adiabatic exponent limit of solutions to the Aw-Rascle traffic model

TL;DR

This paper investigates the formation of delta shock waves in the vanishing adiabatic exponent limit of the Aw-Rascle traffic model, revealing that solutions tend to a delta-shock rather than classical solutions, supported by theoretical proofs and numerical simulations.

Contribution

It provides a rigorous analysis of delta-shock formation in the Aw-Rascle model as the adiabatic exponent approaches zero, extending understanding of traffic flow dynamics.

Findings

Solutions tend to a delta-shock as the adiabatic exponent vanishes.

Riemann solutions with two shocks converge to delta-shocks in the limit.

Numerical simulations confirm the theoretical results.

Abstract

In this paper, we study the phenomenon of concentration and the formation of delta shock wave in vanishing adiabatic exponent limit of Riemann solutions to the Aw-Rascle traffic model. It is proved that as the adiabatic exponent vanishes, the limit of solutions tends to a special delta-shock rather than the classical one to the zero pressure gas dynamics. In order to further study this problem, we consider a perturbed Aw-Rascle model and proceed to investigate the limits of solutions. We rigorously proved that, as $\gamma$ tends to one, any Riemann solution containing two shock waves tends to a delta-shock to the zero pressure gas dynamics in the distribution sense. Moreover, some representative numerical simulations are exhibited to confirm the theoretical analysis.