Suppose that uniform traffic with density up cars per mile approaches the end of a line of traffic stopped at a red ligh

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Suppose That Uniform Traffic With Density Up Cars Per Mile Approaches The End Of A Line Of Traffic Stopped At A Red Ligh 1 (122.6 KiB) Viewed 27 times

Suppose that uniform traffic with density up cars per mile approaches the end of a line of traffic stopped at a red light. Ahead of the light there are no cars, while the stopped traffic is at its maximum density uj cars per mile. At time t = 0, the red light turns green and the front of the line of stopped traffic begins to move forward. One model for the traffic density is Ut + vi(1 – 2u/u1)ux U1 = 0, - < x < 0o, t > 0, Uo if x < -L, լ if - < x < 0 0 if x > 0. u(x, 0) = Assume that the incoming traffic has a density of uo = jul. (a) Show that the characteristics for the initial value problem are given by х х vit + Xo if 0 < x0, -vit + xo if -L < 20 <0, To if x0 <-L. х = Sketch the resulting characteristic diagram. (b) Find a solution u(x, t) of the initial value problem using a combination of shock and rarefaction waves.