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Ford Prefect January 17, 2012 15:52

2nd order (convection) schemes
 
Hello,

I would like to discuss the reasons behind upwind biased 2nd order schemes having higher accuracy than the central difference scheme (while below the Peclet number restriction for the CD scheme). Do the CD scheme become progressively more accurate as we approach a Peclet number of 0? And if so, is the opposite true for the upwind biased schemes?

Cheers!

cfdnewbie January 17, 2012 16:18

Quote:

Originally Posted by Ford Prefect (Post 339795)
Hello,

I would like to discuss the reasons behind upwind biased 2nd order schemes having higher accuracy than the central difference scheme (while below the Peclet number restriction for the CD scheme). Do the CD scheme become progressively more accurate as we approach a Peclet number of 0? And if so, is the opposite true for the upwind biased schemes?

Cheers!

Hello Ford,
I'm wondering: Why/in what way do upwind schemes become more accurate than CD schemes? in the classical FD setting, CD schemes are dissipation-error free, while having larger dispersive errors. Upwind schemes are a lot more dissipative than CD scheme, but have better wave-propagation properties....

The way to analyze the accuracy of a scheme (at least as far as I know) is to actually look at the dissipation and dispersion curves over wavelength...

Cheers!

Ford Prefect January 17, 2012 16:26

Quote:

Originally Posted by cfdnewbie (Post 339799)
Hello Ford,
I'm wondering: Why/in what way do upwind schemes become more accurate than CD schemes? in the classical FD setting, CD schemes are dissipation-error free, while having larger dispersive errors. Upwind schemes are a lot more dissipative than CD scheme, but have better wave-propagation properties....

The way to analyze the accuracy of a scheme (at least as far as I know) is to actually look at the dissipation and dispersion curves over wavelength...

Cheers!

Hey cfdnewbie!

Well I have not made any direct comparisons myself, and admittedly I have only one reference where they compare a QUICK and CD scheme against each other. The QUICK scheme comes out as more accurate of the two compared to the analytical solution. However the reference gave no indication as to why this is the case.

cfdnewbie January 17, 2012 16:41

Quote:

Originally Posted by Ford Prefect (Post 339801)
Hey cfdnewbie!

Well I have not made any direct comparisons myself, and admittedly I have only one reference where they compare a QUICK and CD scheme against each other. The QUICK scheme comes out as more accurate of the two compared to the analytical solution. However the reference gave no indication as to why this is the case.


Hey Ford,
I haven't personally worked with the QUICK scheme, but from Fluent's handbook (which I just googled) it seems that QUICK is a blend of upwind and CD scheme.

Let me take a step back for a second:
In general, the error of a computational scheme is made up of two parts: the dissipative error (which smears out the solution) and the dispersive error (which misjudges frequencies and results in unphysical wiggles). Depending on the leading term in the schemes error, the one or the other error dominates, and both or only one may be present.

Upwind schemes are generally designed to capture wave propagation well, i.e. the show low dispersive errors. On the downside, they are not dissipation-free. (there's always a price to pay for using any form of discretization instead of the real continuous formulation). You could look at this from the other side: Upwind schemes capture waves and introduce dissipation, which is why they work well for convection-dominated problems like shocks.

CD schemes are by design dissipation-error free, but they lack the wave-propagation properties, i.e. they have a dispersive error.

Now, depending on the problem at hand, one or the other type of error may dominate, and one or the other formulation is more accurate. If you have a convection-dominated problem, probably the upwind scheme gives less errors overall, because the flow itself is convection dominated.

If on the other hand you are trying to solve a dissipative problem, you are better of with a CD scheme, where it is essential to capture the dissipation correctly, while wave propagation plays a lesser role.


So saying that one scheme or the other is always better in any kind of flow is a tricky thing to do. Schemes are designed with a certain application in mind.

I hope this helps!

Cheers!

Ford Prefect January 18, 2012 04:53

Hey cfdnewbie,

Tnx for the reply. 2nd order upwind schemes have the same leading term as CD. Perhaps you are thinking of 1st order upwind schemes in your reasoning?
I also stated that we have a flow that is within the CD scheme Peclet number range, i.e. it is a flow with relatively weak convection. Still the QUICK scheme manages to produce a better solution..

Regarding dispersion you could of course choose a TVD scheme that preserves monotonicity, but this is not my original question ;)

Cheers!

cfdnewbie January 18, 2012 07:38

Hey Ford,
sorry, I guess I got carried away a little bit. Well, to answer your original questions, I guess the only good way to find out is to do a dissipation / dispersion analysis of the schemes and compare them....can't think of another way....

sorry again!
cheers!


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