Question Regarding MacCormack Technique
I'm somewhat of a beginner/student to CFD development, and I'm making my way through John D. Anderson's "Computational Fluid Dynamics: The Basics With Applications" wherein Maccormack's finite differencing technique is being utilized to solve the flowfield for a quasi1D, subsonic inflowsupersonic outflow isentropic nozzle. The governing flow equations are reduced for a quasi1D Euler flow, and cast in strong conservation form. In writing my own program to solve this problem, I have been able to match Anderson's results for the first step in time. However, my code becomes unstable as I advance further in time, and I'm unable to obtain a stable solution using the Courant number, grid spacing, and time steps that Anderson specifies.
I'm wondering if this may have something to do with my implementation of calculating the time increment itself, which Anderson is a little vague in how he himself handles this. I know that for each grid point the time increment is calculated, and he in turn picks the minimum time increment calculated across each grid point to advance the solution to the next time step (global time marching approach). However, I'm left wondering if this is repeated for each time iteration, or if this minimum time increment calculated for the first time step is used for subsequent iterations. I have implemented the former approach, and I see that as the solution steps forward in time the minimum time increment calculated across all the grid points diminishes; eventhough, the grid spacing remains constant. Is this normal, or should the time increment remain constant while the solution marches in time? Any insights would be most helpful. Thanks. 
1d Maccormack
Hi Rcktman77,
Recently I was trying to do the same example from Anderson and I experience the same problem. The absolute value of the divergence seems to scale with the timestep but no matter what the timestep is, the solution diverges. Since you encountered this problem a couple of years ago, do you perhaps know what the problem was? With kind regards. Jeroen Wink 
Anderson's CFD book page 336
Jeroen Wink  I think I have stumbled onto the same error. I have coded this thing twice, one in C++ and then again in Excel (first step). The central problem begins on page 336. Once I get the calculations to the end of the predictor step (page 349) , I can't get the same answers for F1, F2, F3 (row 15). The calculations work for row 16 and that's OK. I have tried a couple of things. 1.) Not using primitives in the calculation of J2. 2.) Fixing the d(Area)/dx at rows 0 and 30. I am unsure why he claims one needs the primitives calculated on the top of page 349, because the iteration loops can be closed without that calculation, and then one just extracts the primitives at the end of looping, as needed. The web rumors that a solutions manual exists, maybe we can unravel the mystery from this documentation. I'm not sure how any graduate student could turn in recoded Anderson with out it being somewhat obvious.
Thanks, 
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Hi all,
Apologies for the delayed response. It has been about 5 years since I revisited this problem, so it took me some time to refresh my memory. I looked over my original question, and immediately noted what I originally thought as ambiguous regarding the time step, was in fact stated pretty explicitly in the book. Namely, that global time stepping is used throughout all of the example problems. The time step is calculated at each grid point using Eq. 7.67 and the minimum value from all of the interior points is used for advancing the solution in time (i.e. this value doesn't change once initially calculated). I'm not sure whether this is the same issue you're encountering, but I went back and wrote a program in fortran to solve the problem which I think does so adequately. It doesn't appear to diverge after 1400 iterations at least. I've attached the fortran source for you to review. Please excuse the formatting of the output files. I didn't spend much time formatting them how I intended. Also a lot of the code could probably be refactored to reduce some redundancy, but I felt it was probably best to be as explicit as possible to show the steps as Anderson has written them. Lastly, for whatever reason the forums here don't appear to recognize the f90 extension, so I've renamed my fortran source file using the *.f extension. It's probably best if you rename the attachment prior to compiling using the f90 extension. Please don't hesitate to contact me if you have any additional questions about this problem. Best Regards, Zach 
Thanks a lot for your response.

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Hello,
I have the same problem with my code, the solution always diverges. I did take a look at the FORTRAN code and did try to follow but as i dont know fortran very well maybe i have been making some mistakes. My solution still diverges. Please take a look at my c++ code and see if you can help me out. I have been stuck on this for days. I am at my wit's end.:confused: 
Solved
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Hello again,
So i figured out the the bug. It was a mistake in the calculation of the required flux terms in the corrector step. I have attached the C++ code if someone else faces this problem. 
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hii ...
first of all thanks for the code. but let me ask, the book also describes solving the problem of the subsonicsupersonic nozzle with the nonconservation form of governing equations. However, when I coded the same thing, the solution diverges, and program blew up. Please check if there are possible errors in my code. Thanks a lot. 
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I must apologize for replying this late but i have been a bit busy. After i saw your code, some errors are apparent. You have enforced the the boundary conditions at the inlet and the outlet at the initialization itself. Anderson meant that you should enforce these while solving the flow. You solve for the internal points only and then you extrapolate the value of velocity at inlet and the values of the other primitive floating variables at outlet after each computation. And there is also an error in your solver loop. For the predictor part you are okay but the corrector part needs the predicted value at the inlet at the first step because its uses backward difference. but you havent stored the predicted value of i=0 anywhere. You should run the predictor part on all the internal points separately and then extrapolate predicted values at i=0 and then run the corrected part separately and then finally extrapolate the final values at i=0 and at i=N1. Hope this helps :) 
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Hii..
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thanks a lot.:) 
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