**Math578 - Alexiades**

** LAB 8: 1-D Advection**

Consider a (1-dimensional) advection process, for some concentration **u(x,t)**,
whose initial profile is **u(x,0) = u**_{0}(x), driven by a given (constant) velocity field **V**,
in an (appropriately chosen) interval a ≤ x ≤ b whose ends are impermeable,
during some time 0 <= t <= tmax.
1. State precisely the (full) **mathematical problem** modeling this process.
2. Show that the exact solution of the Initial Value Problem is **u(x,t) = u**_{0}(x−Vt),
assuming **u**_{0}(x) is differentiable.
Take as initial profile the "square bump": u_{0}(x) = 5 for 1 ≤ x ≤ 2, u_{0}(x) = 0 otherwise.
3. Describe what you expect to happen qualitatively and sketch this initial profile u_{0}(x)
and the expected profile at time t=4 if V=1 and if V=5. How far does the pulse travel ?
4. Implement the explicit upwind scheme for this problem,
and derive the CFL stability condition.
In your code, use: Dt_{expl} = Dx/V , Dt = factor*Dt_{expl}, so factor can be changed easily
(make it an input parameter along with V, a, b).
Assume constant velocity V>0, and use: MM = 32, tmax=4.0, dtout=2.0,
a=0, and b big enough so that the entire action happens inside [a,b] (from 3.).
Every dtout (starting at time=0.), your code should output the time, number of time-steps,
maximum U-value, and also the entire profile of U (including at time tmax) for plotting.
For each of the cases listed bellow, do the following:

plot the initial and final profiles on one plot ( *set yrange [0:5.5]* in gnuplot 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
(annotations can be by hand on the plot printouts).
look at the {time,Umax} pairs you generated, comment on what you observe.
5. Here are the cases to examine.
(a) V = 1., factor = 1.0
(b) V = 5., factor = 1.0
(c) V = 5., factor = 1.05
(d) V = 5., factor = 0.9
(e) V = 5., factor = 0.5