Initial condition is:
The velocity is 1.0 m/s. The length of pipe section being
modeled is 5 meters, and end time for the transient is 2.5
seconds.
Part A
For both explicit and implicit, donor cell (first order upwind)
difference methods on a uniform ten volume mesh,
make three runs
with Courant numbers of 0.01, 0.5, and
1.0. On a single graph, plot
the density vs. x for the analytic solution and all six runs at 2.5
seconds. Note the difference in behavior between explicit
and implicit methods. Which has a higher numerical diffusion?
Part B
Now extend your analysis to generate results with Leith, QUICKEST (both are explicit), Implicit QUICK, and Explicit Quick methods. Again use a uniform 10 volume mesh. For Leith, QUICK, and QUICKEST use a zero-curvature exit boundary condition (density beyond right boundary is a linear extrapolation from the last two densities on the mesh). Graph the results (density vs. spatial location) at t=2.5s from these four methods plus the two 1st Order Donor Cell methods on one plot for a material Courant number of one. Make a second such plot for a material Courant number of 0.5. Discuss what you observe.Now rerun the six methods with 100 uniform spatial mesh cells
along
the 5 meter section of pipe ( cell length = 0.05 m). This time
only make runs for a material Courant number of 0.5.
Convince yourself that Explicit QUICK is unstable, then graph the
density profiles at 2.5 seconds for all methods except Explicit QUICK
on
the same plot. In addition make one plot for each method comparing the
results
for material Courant number of 0.5 at the two mesh sizes.
Label the curves clearly.
Part C
Run Richardson based error analysis on implicit QUICK and
explicit QUICKEST to check the effective order of accuracy in time and
space for these methods. Use about 180 volumes as the finest mesh
and
use 0.001s as the smallest time step in the two studies.
You will in theory be able to calculate values of the order "p" at 20
spatial locations, but remember that not all locations may produce
valid results. For each method report your results to me as two
plots (one for time and one for space order) of valid values of p as a
function of spatial location.
In addition to the plots provide me with written equations descibing each difference method used for the advection equation.
Remember to submit your program.
If you need to check your results try modifying this program to solve the homework problem. At the moment my sample uses a simpler set of boundary conditions.