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Old   July 20, 2009, 15:49
Question About the central differencing scheme
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Hi, CFD experts:

I have a simple question to ask. I am reading Versteeg and Malalasekra's book "An Introduction to Computational Fluid Dynamics: The Finite Volume Method," it says that the central differencing scheme for convection-diffusion problems is not a suitable discretisation practice for general-purpose flow calculations since it will be stable and accurate only if Pe<2. And it does not posesss the transportiveness at high Pe and violate the boundedness also.

But when I read Ferziger and Peric's book "Computational Methods for Fluid Dynamics," it doesn't mention CDS's weaknesses. From this book's chapter 3.11, it seems that CDS is a better choice, compared to upwind scheme. I am confused. Which view is right? Thank you in advance!

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Old   July 21, 2009, 03:34
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the FUD is only first order accuracy , but it is absolute stablized. the CD scheme is second order accuracy , but it is conditonnable stabilized.
for the problem in the practice, convection will dominant the diffusion proce.. .FUD can not give accurate results. So Cd is more prefered.
CD is conditonnable stabilized, we also have other absolute high order schemes such as MUSCL, SMART, etc ,which are all absolute stablized and have a high accuracy.
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Old   July 21, 2009, 04:00
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Quote:
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Hi, CFD experts:
...
But when I read Ferziger and Peric's book "Computational Methods for Fluid Dynamics," it doesn't mention CDS's weaknesses.
...
It does. Really.

However, F&P recommend to blend the second order accurate Central Differencing Scheme (CDS) with the first order Upwind Differencing Scheme (UDS). Only "a small amount of blending" is "normally" needed and a variable called 'gamma' is used for the blending. A good target is to use, say, in the order of, 80% CDS and 20% UDS. This fixed blending has the advantage that you'll be sure that 80% CDS is also used in those places where dynamic blending would bring the percentage down to full UDS (albeit with the risk of (locally) unbounded results).
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Old   July 21, 2009, 07:47
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Quote:
Originally Posted by Hendrik View Post
It does. Really.

However, F&P recommend to blend the second order accurate Central Differencing Scheme (CDS) with the first order Upwind Differencing Scheme (UDS). Only "a small amount of blending" is "normally" needed and a variable called 'gamma' is used for the blending. A good target is to use, say, in the order of, 80% CDS and 20% UDS. This fixed blending has the advantage that you'll be sure that 80% CDS is also used in those places where dynamic blending would bring the percentage down to full UDS (albeit with the risk of (locally) unbounded results).
Hello:
Hendrik, i am glad to see you here. I know you, i often go to the dolfn forum.
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Old   July 21, 2009, 10:44
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ztdep and Hendrik, thank you vey much for your useful information! Based on Hendrik's answer, I have another question. Where did F&P talk about blending of CDS and UDS in their book? Thanks!
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Old   July 21, 2009, 16:39
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Amongst other places: Chapter 3.11 Example and 4.7 Examples with "... CDS ... but it oscillates. Local Grid refinement would help localize and, perhaps even remove, the oscillations as will be discussed in Chap. 11. The oscillations could also be removed by locally introducing numerical diffusion (e.g. by blending CDS with UDS)."

Of course it also depends on the quality of your mesh (cell shapes, non-orthogonalties, jumps etc.)
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Old   July 21, 2009, 17:35
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Thank you for your reply!
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