Tuesday, May 9, 2017

Case: initial density decreasing downstream:

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:

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:

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

Jump condition:

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

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: