Flux Perturbation And Interactions Of Classical Waves To The Aw-Rascle Traffic Flow Model

The Aw-Rascle model(AR model)is an important macroscopic traffic flow model,which has been widely used to investigate practical traffic problems such as the formation of traffic jams.At present,many scholars have studied the formation of delta shock wave and vacuum for AR model and discussed the wave interactions.However,most of these studies involve only pressure perturbation rather than the more general flux perturbation.As far as the model is concerned,it mainly focuses on the homogeneous case,while the discussion on the inhomogeneous one is relatively few.In view of this,on the one hand,this dissertation solves flux-perturbation problem of the homogeneous AR model and explores the formation of delta shock wave and vacuum in vanishing flux-perturbation limits.On the other hand,the time-dependent general external force is introduced to study the flux-perturbation problem and interactions of classical waves for the inhomogeneous AR model.First,the flux-perturbation problem of the homogeneous AR model with generalized Chaplygin gas pressure is solved.With the help of some classical analytical techniques,the uniqueness of delta-shock solution with flux-perturbation parameter is strictly proved.It is found that,as both the flux perturbation and pressure vanish,the Riemann solution of the perturbed system containing a shock wave and a contact discontinuity first tends to some delta shock wave of the flux-perturbation system itself,and only then to the delta-shock solution of the zero-pressure flow.This formation mechanism of delta shock wave is obviously different from that of the case with only pressure perturbation or the state equation is barotropic gas.Besides,we prove that the Riemann solution containing a rarefaction wave and a contact discontinuity converges to the vacuum solution of the zero-pressure flow.Numerical simulation results consist with theoretical analysis.Second,the flux-perturbation problem of the inhomogeneous AR model with source term is discussed.We develop a special variable substitution to rewrite the inhomogeneous system into a conservative one,and three different structures of solution are constructed.The results show that due to the influence of the external force,the Riemannian solution of the inhomogeneous AR model is no longer selfsimilar,and all the wave curves are no longer straight lines,which is essentially different from the homogeneous case.Furthermore,it is proved that as the pressure and flux approximation vanish,the solution of the flux-perturbation system containing a shock wave and a contact discontinuity first tends to some critical delta shock wave of the inhomogeneous system itself,and only then to the delta shock wave of the inhomogeneous zero-pressure;while the vacuum solution of the inhomogeneous zero-pressure flow is the limit of Riemann solution of the flux-perturbation system containing a rarefaction wave and a contact discontinuity.Numerical results confirm the theoretical analysis.Finally,the extended Chaplygin gas pressure is borrowed to study the interactions of classical waves for the inhomogeneous AR model.Two different types of Riemann solutions involving rarefaction wave,shock wave and contact discontinuity are constructed by the method of analyzing in phase plane.Then,via discussing the collision and overtaking of waves,the global structures of solutions for the Riemann problem with initial data of three piecewise are constructed and the stability of which under the small perturbation of initial data is briefly discussed.