Traffic Flow Analysis Of The Ar Model

With the development of society , the study of traffic flows becomes more and more important . For this purpose , here are many mathematical traffic flow models ,one of which is the AR model .We devide the paper into three parts .The first part , we simply introduce the following models : LWR model , PW model , Zhang model , and AR model .The second part , we will analysis the AR model indetail .By analyzing the structure , we can get three basic waves : the centered rarefaction wave , the shock wave and the contact discontinuity . Then we define the Riemann invariants by the knowledge of the hyperbolic conservation law . Secondly we construct the Riemann solution on the (ρ, v) plane . Finally we can get the existence of the weak solution by the Glimm method . The Glimm scheme is very important in solving the Cauchy problem for the systems of conservation laws . It is difficult to get the bounded total variation of the solutions , so it is hard to get the existence of the weak solution . However for the AR model , we will find that the centered rarefaction wave curve , the shock wave and the contact discontinuity are lines in the Riemann invariant plane , and the total variation of solution is descreasing in time . So we prove that the solution is bounded and has a bounded total variation .In the third part , we simply compute the AR model by using the Lax-Friedrichs scheme .