Day 26

Tuesday, May 9, 2017

A model of traffic flow

2017_05_09/20170509_133917.jpg 2017_05_09/20170509_133939.jpg 2017_05_09/20170509_133948a.jpg

Example

Case: initial density decreasing downstream:

traffic_characteristics_1.png

We can easily calculate the density at any place or time because the density is constant along the characteristics.

Case: some initial density increase downstream:

traffic_characteristics_2.png

After some time (t=1 in this case), an inconsistency arises: characteristics collide, giving inconsistent information about u at the point of collision.

Here is another view:

shock_forms.png

Beyond t=1 we have nonsensical multiple values of u predicted for some values of x.

Shocks (discontinuity of u along a curve)

2017_05_09/20170509_133948b.jpg

Jump condition:

2017_05_09/20170509_133954.jpg

A population age-profile model

2017_05_09/20170509_135819.jpg

Here is an animation (If not moving, try Firefox.):

allpop1.png

If we scale the population at each moment of time so that the maxiumum is always 1, we can see how the shape of the population age profile settles to steady state:

allpop1norm.png