Concentration of mass in the vanishing adiabatic exponent limit of Aw–Rascle traffic model with relaxation

In this paper, we study the occurrence mechanism on the phenomenon of concentration and the formation of delta-shock wave in vanishing adiabatic exponent limit of Riemann solutions to the Aw–Rascle model of traffic flow with a relaxation term. At first, by a special type of variable substitution, the Riemann problems of pressureless gas dynamics and Aw–Rascle model of traffic flow with a relaxation term are solved. Unlike the homogeneous case, both Riemann solutions are no longer self-similar due to the relaxation term. Then, we rigorously prove that, as the adiabatic exponent vanishes, the Riemann solution containing a shock curve and a contact discontinuity of the Aw–Rascle model with a relaxation term tends to a special delta-shock wave, which is different from the delta-shock wave of limiting pressureless gas dynamics model. At last, some numerical simulations are presented to show the formation process of delta-shock waves and illustrate the above analysis.