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

Article type: Research Article

Authors: Sheng, Shouqiong | Shao, Zhiqiang^{; *}

Affiliations: College of Mathematics and Computer Science, Fuzhou University, Fuzhou 350108, China

Correspondence: [*] Corresponding author. Tel.: +86-0591-83852790; E-mail: zqshao@fzu.edu.cn.

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 the adiabatic exponent 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.

Keywords: Aw–Rascle traffic model, Riemann solutions, delta shock wave, vanishing adiabatic exponent limit, zero pressure gas dynamics, weighted Dirac-measure, numerical simulations

DOI: 10.3233/ASY-211725

Journal: Asymptotic Analysis, vol. 129, no. 2, pp. 179-213, 2022