[yhy20081016]

Hi, everybody.
Recently I read the AIAA "Editorial Policy Statement on Numerical and Experimental Accuracy".
(http://arc.aiaa.org/page/numericalan...mentalaccuracy)
This guide requires that numerical algorithm solving partial differential equations should be at least formally second-order accurate in space.
I know that this is because first order schemes will introduce too much numerical dissipation. However, numerical dissipation can be reduced by refining the grid. Therefore, I have a question. If I use first order scheme and obtained a grid-independent solution by successively refining the grid, then is the solution acceptable? (For my problem, high order schemes often make the solution process unstable, and even diverge. ) Thank you very much.

[duri]

This policy related question, AIAA editors are best people to answer this question. But from practical point of view, no one uses first order results for their application. Also, there is no point is working on first order with bulky mesh. It will increase unnecessary computational resource and time. If you are concerned about stability of high order scheme, then please post what are you trying to do.

[FMDenaro]

using first order upwind for generally used high Re number flows,
1) is not really suitable to reach a grid independent solution in terms of DNS
2) is not usable for LES as it imply a dramatic smoothing of the resolved frequency
3) can be some how acceptable in RANS as long as the turbulence modelling overcome the first order magnitude of the local truncation error.

In conclusion, very low Re laminar flows can be resolved but are often academic test-cases.

High-order schemes are accurate and stable (conditionally), if you have a numerical instability something does not work correctly.

[yhy20081016]

Originally Posted by FMDenaro

using first order upwind for generally used high Re number flows,
1) is not really suitable to reach a grid independent solution in terms of DNS
2) is not usable for LES as it imply a dramatic smoothing of the resolved frequency
3) can be some how acceptable in RANS as long as the turbulence modelling overcome the first order magnitude of the local truncation error.

In conclusion, very low Re laminar flows can be resolved but are often academic test-cases.

High-order schemes are accurate and stable (conditionally), if you have a numerical instability something does not work correctly.

You said that first order upwind scheme is "some how acceptable" in RANS. Which of the following do you refer to?
(1) Use first-order upwind scheme for both the momentum equations and the turbulence transport equations;
(2) Use high-order schemes for the momentum equations, while use first-order upwind scheme for the turbulence transport equations.

[julien.decharentenay]

Originally Posted by duri

But from practical point of view, no one uses first order results for their application.

While I do not discuss the point that Duri is trying to make, the above sentence would gaiin from being more accurately defined.

no one: the statement is obviously an expression of speech and overgeneralisation;

uses first order results for their application: if you are using a commercial software, you are likely to be using a spatial discretisation scheme that is not strictly second-order. It is more likely to be a second-order scheme with (a) a limiter, and (b) a blending with a first order. Furthermore, the second-order is achieved by an explicit correction. The implicit part is very likely to be first order only.

I would personally reformulate the statement as follows: Best practice would recommend the use of second-order spatial discretisation scheme.