Consider a (1-dimensional) advection-diffusion process, with (constant) diffusivity

whose initial profile is

whose ends are impermeable, during some time

1. State precisely the full

Take as initial profile the "square bump": u

2. Describe what you expect to happen qualitatively and sketch (by hand) the initial profile

expected profile at time

3. Implement the explicit upwind scheme + diffusion for this problem, and

In your code insert a "factor" in the time-step, as before.

Assume constant velocity

Choose

Every

and the entire profile of U (including at time tmax) for plotting.

Now, we want to examine how the presence of diffusion affects the advection profiles obtained in Lab 8.

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

and make comments/observations as to what you think is happening, how it compares with other cases and why.

The

Here are the cases to examine.

4. Advection-Diffusion: Low velocity:

(4a) V=1., D=0.0, factor=1.0

(4b) V=1., D=0.5, factor=1.0

(4c) V=1., D=0.1, factor=1.0

(4d) V=1., D=0.1, factor=0.9

(4e) V=0., D=0.1, factor=0.9

Which of (4c) , (4d) is more accurate?

5. Advection-Diffusion: High velocity:

(5a) V=5., D=0.0, factor=1.0

(5b) V=5., D=0.5, factor=1.0

(5c) V=5., D=2.5, factor=1.0

6. Do the cases with same

7. Implement Super-Time-Stepping and examine the effect in cases (5b) and (5c), with MM=128.

To compute error, take Euler solution (N=1, ν=0) as U_exact. Try Nsts=10 or 20, and some (small) ν.

Is there any speedup? (Speedup = nsteps_Euler / nateps_STS)