Advection Equation

Start with a simple advection equation assuming a constant positive velocity v:

The boundary condition is:

Boundary Condition

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.