Math578 - Alexiades
                  LAB 8:   Advection - Diffusion

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Consider a (1-dimensional) advection-diffusion process, with (constant) diffusivity D > 0, for some concentration u(x,t),
whose initial profile is u(x,0) = u₀(x), driven by a given velocity field V, in an interval a ≤ x ≤ b (to be chosen appropriately, see below).
whose ends are impermeable, during some time 0 ≤ t ≤ t_end.

1. State precisely the full mathematical problem modeling this process (PDE, where it holds, initial condition, boundary conditions).

Take as initial profile the "square bump": u₀(x) = 5 for 1 ≤ x ≤ 2, u₀(x) = 0 otherwise.

2. Describe what you expect to happen qualitatively and sketch (by hand) the initial profile u₀(x) and the
  expected profile at time t=4 if V=1 and if V=5 (and D =0 or small). How far does the pulse travel ?

3.a. Implement the explicit upwind scheme + diffusion for this problem,
    In your code insert a "factor" in the time-step, as before.
  b. derive the CFL stability condition.

Assume constant velocity V > 0 and diffusivity D > 0, and take MM = 32 , tend=4.0 , dtout=2.
  Choose a and b appropriately, so that the entire action happens inside [a,b].

Every dtout (starting at time=0), your code should output the time, number of time-steps, maximum U-value,
  and the entire profile of U (including at time t_end ) for plotting.

Now, we want to examine how the presence of advection affects the diffusion profiles obtained in Lab2.

For each of the cases listed bellow, do the following:

Plot the profiles at times 0, dtout, tend, on one plot  ( set yrange [0:5.5] to see the top clearly ).

On the plot, mark the parameter values that generated it, mark which curve is at what time,
  and make comments/observations as to what you think is happening, how it compares with other cases and why.

Look at the {time, Umax} pairs you generated, comment on what you observe.

Here are the cases to examine.

4. Pure advection:
      (4a) V = 1. , factor = 1.0
      (4b) V = 5. , factor = 1.0
      (4c) V = 5. , factor = 1.05
      (4d) V = 5. , factor = 0.9
      (4e) V = 5. , factor = 0.5

5. Advection-Diffusion: Low velocity:
      (5a) V=1., D=0.0, factor=1.0   (pure advection)
      (5b) V=1., D=0.5, factor=1.0
      (5c) V=1., D=0.1, factor=1.0
      (5d) V=1., D=0.1, factor=0.9
      (5e) V=0., D=0.1, factor=0.9   (pure diffusion)
  Which of (5c) , (5d) do you think is more accurate? Why?

6. Advection-Diffusion: High velocity:
      (6a) V=5., D=0.0, factor=1.0
      (6b) V=5., D=0.5, factor=1.0
      (6c) V=5., D=2.5, factor=1.0

Submit on Canvas:   answers, plots, comments in "Lab8.pdf" , and your code in "Lab8-code.txt".