# First order in time and Central Difference Convergence problem

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 February 12, 2010, 14:40 First order in time and Central Difference Convergence problem #1 Member   Ramesh K Join Date: Dec 2009 Location: Bangalore Posts: 65 Rep Power: 10 Hi all I am trying to solve compressible flow of gas in a cylinder using N-S equations. I have written a code using finite volume method with first order in time and central difference in space. My grid is generated by cylindrical coordinates and the solver in cartesian(meaning grid has concentric circles and radial lines)... I have problem in convergence... the solution oscillates and diverges... I have added artificial dissipation also then I see these oscillations and divergence other wise it only diverges I have tried with various cfl ranging from 0.9 to 0.0001 but failed to get convergence can any body tell what exactly the problem might be.... I have checked the code to see the total flux in the initial condition for all the cells to be zero and ensured proper application of boundary conditions.. any suggestions of what can be checked

July 17, 2010, 07:52
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Akinola
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 Originally Posted by RameshK Hi all I am trying to solve compressible flow of gas in a cylinder using N-S equations. I have written a code using finite volume method with first order in time and central difference in space. My grid is generated by cylindrical coordinates and the solver in cartesian(meaning grid has concentric circles and radial lines)... I have problem in convergence... the solution oscillates and diverges... I have added artificial dissipation also then I see these oscillations and divergence other wise it only diverges I have tried with various cfl ranging from 0.9 to 0.0001 but failed to get convergence can any body tell what exactly the problem might be.... I have checked the code to see the total flux in the initial condition for all the cells to be zero and ensured proper application of boundary conditions.. any suggestions of what can be checked

I am having the exact same problem now. How did you go about resolving it? i will be very grateful for your reply. Thanks

 July 17, 2010, 08:09 #3 Super Moderator     Praveen. C Join Date: Mar 2009 Location: Bangalore Posts: 277 Blog Entries: 6 Rep Power: 11 First of all, I would recommend trying a higher order time integration scheme since the first order scheme coupled with the high order space discretization is generally unstable. The 3-stage TVDRK scheme of Shu and Osher or the standard 4-stage RK scheme would be good choices.

July 17, 2010, 08:35
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 Originally Posted by praveen First of all, I would recommend trying a higher order time integration scheme since the first order scheme coupled with the high order space discretization is generally unstable. The 3-stage TVDRK scheme of Shu and Osher or the standard 4-stage RK scheme would be good choices.
I thought of that as well but I need to apply implicit scheme because the process I am trying to simulate can have physical time of up to 2 hours and I need to be able to use as high a time step as possible. I am trying to avoid the stringent CFL condition if you guess what I mean. From my lit review, only the first order Euler time integration is unconditional stable. I hope you get my point?

July 17, 2010, 08:39
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Praveen. C
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 Originally Posted by falopsy I thought of that as well but I need to apply implicit scheme because the process I am trying to simulate can have physical time of up to 2 hours and I need to be able to use as high a time step as possible. I am trying to avoid the stringent CFL condition if you guess what I mean. From my lit review, only the first order Euler time integration is unconditional stable. I hope you get my point?
My response was to RameshK. I am assuming he is using explicit, first order in time scheme.

July 17, 2010, 08:49
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 Originally Posted by falopsy I thought of that as well but I need to apply implicit scheme because the process I am trying to simulate can have physical time of up to 2 hours and I need to be able to use as high a time step as possible. I am trying to avoid the stringent CFL condition if you guess what I mean. From my lit review, only the first order Euler time integration is unconditional stable. I hope you get my point?
Actually you can find second and high order schemes for the time integration, implicit schemes and also uncoditional stable (see for example Ferziger and Peric) or you can search for the dual time step method of Jameson. The last one is the method of choice in a few commercial CFD codes.

Do

July 17, 2010, 09:06
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 Originally Posted by praveen My response was to RameshK. I am assuming he is using explicit, first order in time scheme.
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 Originally Posted by DoHander Actually you can find second and high order schemes for the time integration, implicit schemes and also uncoditional stable (see for example Ferziger and Peric) or you can search for the dual time step method of Jameson. The last one is the method of choice in a few commercial CFD codes. Do

@Praveen, I know you are referring to RameshK but I am facing similar issues and will appreciate any help.

@DoHander; will try to implement the other methods but will like to debug my program with a simple scheme cos it will be easier to detect bugs in them. I am guessing my problem is not from the time discretisation because the solution did not even converge for the first time step. I am looking at the first step of SIMPLE scheme ie finding velocity with a fixed pressure fields. I hope you can help me. Thanks

 July 17, 2010, 14:13 Bug in the program #8 Member   Ramesh K Join Date: Dec 2009 Location: Bangalore Posts: 65 Rep Power: 10 Hi Actually there was a bug in calculating the velocity magnitude in the code, fixed it and got the desired convergence. I have solved the same problem for pressure ratio (Po/P)of 1.2 and 1.1 where Po is total pressure at inlet and static pressure at exit, used RK 4 for time integration and solved it. But for convergence I have worked both Euler and Ns equations, for euler the convergnece went all the way to 1e-16 but for NS it was oscillating at 1e-4. NS code I have run for nearly 3 lakh time steps.

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