Approaching Chaplygin pressure limit of solutions to the Aw-Rascle model

This paper studies the limit of solutions to the Aw-Rascle model as the pressure tends to the Chaplygin gas pressure. For concreteness, the pressure is taken as a modified Chaplygin gas pressure. Firstly, the Riemann problem for the Aw-Rascle model with the modified Chaplygin gas pressure is solved constructively. Secondly, it is shown that as the pressure tends to the Chaplygin gas pressure, some Riemann solutions containing a shock and a contact discontinuity tend to a delta-shock solution, whose propagation speed and strength are different from those of delta-shock solution to the Aw-Rascle model with a Chaplygin gas pressure. Besides, it is also proven that the rest Riemann solutions converge to a two-contact-discontinuity solution, which is exactly the solution to the Aw-Rascle model with a Chaplygin gas pressure. Thirdly, some numerical results are presented to exhibit the process of formation of delta-shocks.