This is my HW4 web page, it contains animations and links to my report in PDF and HTML versions. The problem description is here PDF which contains detailed description of the 3 problems solved.
The following is my solution report in HTML and PDF
The following are animated GIFs showing the ﬁnite diﬀerence numerical solution to problem 1 as described in the above HW. The scheme used is LaxWendroﬀ.
Clicking on an image will start the animation in a new window.
These simulations only show the pressure wave, and not the acoustic perturbation velocity .
This solution was run with boundary conditions which caused the sound wave to reﬂect from both boundaries. This is what would happen inside a room with reﬂective walls such as concerete or wood.
This solution was run with boundary conditions which caused the sound wave to reﬂect from only the left boundary but absorbed into the right boundary. This is what would happen inside a room with one wall treated with material to absorbe the sound waves reaching it.
The following are animations of the numerical solution to . The solution used the ﬁnite volume method using 7 diﬀerent numerical ﬂux limiter functions to compare performance.
These 7 methods are deﬁned in the problem statement in the report above.
The methods are
The following tables show the results of the simulations. 4 tables are given. Each table is for a diﬀerent initial conditions. In all of these results, the maximum run time was seconds. In order to reduce the size of the animation ﬁle, not every frame was captured from the simulation run.
Courant number used was , the advection speed was set at and grid spacing was meters. The domain is using cell centered grid.
These animations will run only once and stop at 5 seconds. To run it again, simply reload the web page using the browser reload button, this will cause the animation to start from the beginning again.
The wave packet is deﬁned as
 


Upwinding: Large diﬀusion at wave crest and trough. but no shift. Grade 
LaxWendroﬀ: Some diﬀusion at wave crest and trough, in addition of signiﬁcant shift to the left relative to exact solution, grade: 
BeamWarming: Similar to LaxWendroﬀ, but shift is to the right. Grade 
Minmod: diﬀusion at wave crest and trough, as with upwinding, but not as bad. No shifting. Grade



Superbee: NO shifting, and very small amount of diﬀusion at the crest and trough. Grade 
MC Limited: Similar to superbee, but a little more diﬀusion at the crest and trough. Grade 
Van Leer: Similar to MC limited, but even more diﬀusion at the crest and trough. Grade 

The wave packet is deﬁned as
 


Upwinding: No shifting, but large amount of diﬀusion at the crest and trough of the wave. Grade 
LaxWendroﬀ: Did well, no shifting nor diﬀusion seen. Grade 
BeamWarming: Similar to LaxWendroﬀ. Grade 
Minmod: No shifting, but small amount of diﬀusion near the crest and trough. Grade



Superbee: No shift and no diﬀusion. But at the crest and trough, the numerical solution appeared to less smooth than with LaxWendroﬀ making a small straight edge. Grade 
MC Limited: Similar to LaxWendroﬀ, better than Superbee around the crest and trough, no straight edge appeared. Grade 
Van Leer: No diﬀusion and no shifting. Grade 

A step function from to .
 


Upwinding: No ripples seen, follows the general form of the step function, but large amount of diﬀusion near the corners. Grade 
LaxWendroﬀ: Large ripples around the corners on the left of the step function. Less diﬀusion than upwinding. Grade 
BeamWarming: The ripples are larger and have a larger extent than LaxWendroﬀ. Grade 
Minmod: No ripples and little diﬀusion. An improved version of upwinding. Grade



Superbee: The best scheme for the step function. No ripples, very closely followed the exact solution, but very small diﬀusion is still there. Grade 
MC Limited: Similar to supperbee, but more diﬀusion. Grade 
Van Leer: Similar to MC limited. Grade 

The initial condition used for this test is
This test just combines the step function with the low frequency smooth test done above. Hence, the same comments will apply as above.