NDSolve for PDE with discontinuous initial/terminal condition

Question

I have an issue with NDSolve for the case of a PDE with discontinuous initial/terminal condition. Consider the PDE solution

Z=z/.First[NDSolve[{
  D[z[t,x],t] == z[t,x] D[z[t,x],x]-1/2 (1-t)^2 D[z[t,x],{x,2}],
  z[1,x]==UnitStep[x],
  z[t,-5000]==0,
  z[t,5000]==1},
  {z},{t,0,1},{x,-5000,5000}]]

The terminal condition is the discontinuous UnitStep function. When I plot the solution for t=1 now,

Plot[Z[1, x], {x, -10, 10}]

Mathematica has obviously smoothened the terminal condition (which was explicitly given by the UnitStep function).

Is it possible to disable this smoothing? Or at least, I want to have a smooth function that does not exceed 1 and does not go below 0.

Note: What I am not searching for is a solution like "increase the MaxSteps" or something similar. I am interested in options on how Mathematica processes the initial/terminal condition.

Thanks a lot for your help!!

Answer 1

For the given grid resolution, you're probably going to have to smooth the UnitStep function with a width equal to that grid spacing (at least). Here is one way to do that, using the Fermi function instead of UnitStep:

f[x_, d_] := 1/(E^(-(x/d)) + 1)
Z = z /. First[
   With[{d = 5000./10000}, 
    NDSolve[{D[z[t, x], t] == 
       z[t, x]*D[z[t, x], x] - 1/2*(1 - t)^2*D[D[z[t, x], x], x], 
      z[1, x] == f[x, d], z[t, -5000] == f[-5000, d], 
      z[t, 5000] == f[5000, d]}, {z}, {t, 0, 1}, {x, -5000, 5000}]]]

Plot[Z[1, x], {x, -10, 10}]

Here, the parameter d is the smoothing of the vertical step.