Dear all,

How superior is the QUICK scheme to 2nd order upwind scheme in terms of numerical diffusion???

i often use quick scheme as high order scheme or powerlaw scheme as low scheme i think 2nd order upwind has a strange converge charactericstic

QUICK is a second order scheme and it upwinds. It is the obvious way to write a conservative, second order upwind scheme and so I do not recognise "2nd order upwind scheme". What is it?

If you wish to answer to your question, perform a Taylor Series Expansion and examine the coefficients of the error terms.

I think quick scheme is better than 2nd ordr upwind for the most fluid cases!

I just finished looking at the dissipation/dispersion characteristics of a variety of algorithms. QUICK is superior to 2nd order upwind - it damps the high frequency components less and has less dispersion as well. QUICK actually is slightly better than MUSCL with regard to damping, although MUSCL has less dispersion. This is true for first or second order time differencing.

Where can I find a book that discusses all of these issues from the point of view of various schemes?

Most any basic CFD book will have a chapter on Fourier analysis of difference schemes to determine dissipation and damping. You can also find a lot of information on the Internet if you search for damping and dispersion in CFD. Your best bet is to learn how to apply the analysis techniques (it's not that hard) and then use them for the various schemes you are interested in.

The Quick scheme is a third order scheme (not a second order), it has stability problems. For full details check the following book Computational Fluid Mechanics, an introduction for engineers by MA Abbott and DR Basco published by Longman Scientific & Technical

No. It is a second order scheme. It has no particular stability problems relative to similar schemes so long as you are careful about the implementation next to boundaries. However, it is unbounded which can cause problems for physical quantities which should not become, for example, negative.

When it was originally published Leonard tried to talk up its accuracy with some doubtful hand waving about polynomials (it drops an order of accuracy for the size of the computational molecule inorder to get conservation). This has been a source of confusion. The acid test is to perform a grid refinement study and look at the slope. It will be second order.

As far as I know, the QUICK scheme is third order accurate when the grid is uniform and is second order accurate when the numerical grid is nonuniform. In FVM, even though the grid is uniform, the QUICK scheme is second order accurate since if you want to evaluate the value at west control volume face (w), you need the values at points P. W, WW. and the distances between P-w, w-W, W-WW are not uniform. However, the QUICK scheme is third order accurate if one uses the FDM with uniform grid.

No. It takes about 2 minutes to perform a Taylor Series Expansion and get the leading error term as (dx^2)/24 f''' for df/dx on a uniform grid.

Visit the following website

http://widget.ecn.purdue.edu/%7Ejmurthy/me608/main.pdf

and read 113 page, 5.6.2 Third Order Upwind Schemes.

The derivation is incomplete because it considers only a single flux. In order to determine the accuracy of a scheme you need to consider a cell. This can be done in a few minutes using a Taylor series expnansion.

Andy

I would be a good idea if you check the reference I gave you before, the scheme is well developed there and the stability diagram is published too. Good luck with whatever you are doing

Why? What does it add that is not in Leonard's papers? Or a simple Taylor series expansion or stability analysis?

I did not mind pointing out the "error" in Halim's reference (or more his interpretation of the order of a flux instead of the order of scheme) because it was online and Purdue are a good group. But I am not going to waste my time ordering an introductory CFD book from the library just to point out possible errors in it.

[santiagomarquezd]

Quoting Hirsch (Numerical Computation of Internal and External Flow)

Comment on the Quick scheme

There is some controversy in the literature, as to the claim of the third order accuracy of the Quick scheme. As seen from Problem P.4.17, the formula in the middle column of the above table is indeed a third order approximation for the mid-cell value ui + 1/2 , as is shown by comparing with the Taylor expansion of ui + 1/2 around ui . However, when considered as a formula for the first derivative based solely on the mesh point values, and applying Taylor expansions of the points ui − 2 , ui − 1 , ui + 1 , around ui , the formula for the first order derivative shown in the last column of table 8.4.2 is only second order accurate. Referring to Problem P.4.13 in Chapter 4, the finite difference formula for the first derivative is of third order for the parameter a = 1/6; while the Quick scheme corresponds to a = 1/8, leading to a dominating truncation error equal to −1/8 x2 · uxxx .
On the other hand, if we would work with cell-face values ui + 1/2 and ui − 1/2 as basic variables, then the Quick approximation would indeed lead to third order accuracy. However, this is rarely the case in practice, where in many codes the mesh point variables or the cell-averaged values are the reference quantities.

Regards

[randolph]

Originally Posted by Halim Choi
;35452

As far as I know, the QUICK scheme is third order accurate when the grid is uniform and is second order accurate when the numerical grid is nonuniform. In FVM, even though the grid is uniform, the QUICK scheme is second order accurate since if you want to evaluate the value at west control volume face (w), you need the values at points P. W, WW. and the distances between P-w, w-W, W-WW are not uniform. However, the QUICK scheme is third order accurate if one uses the FDM with uniform grid.

Quick is third order accurate by construction.

The reason the why QUICK does not show third order accurate under FVM is because most of FVM code is second order accurate (people don't differentiate cell value and cell averaged value). That's why QUICK does not show third order accuracy in FVM. However, if you evaluate QUICK in FDM, it will show its third order accuracy

[randolph]

Originally Posted by andy
;35449

No. It is a second order scheme. It has no particular stability problems relative to similar schemes so long as you are careful about the implementation next to boundaries. However, it is unbounded which can cause problems for physical quantities which should not become, for example, negative.

When it was originally published Leonard tried to talk up its accuracy with some doubtful hand waving about polynomials (it drops an order of accuracy for the size of the computational molecule inorder to get conservation). This has been a source of confusion. The acid test is to perform a grid refinement study and look at the slope. It will be second order.

Quick is third order accurate by construction.

The reason the why QUICK does not show third order accurate under FVM is because most of FVM code is second order accurate (people don't differentiate cell value and cell averaged value). That's why QUICK does not show third order accuracy in FVM. However, if you evaluate QUICK in FDM, it will show its third order accuracy

[FMDenaro]

Originally Posted by randolph

Quick is third order accurate by construction.

The reason the why QUICK does not show third order accurate under FVM is because most of FVM code is second order accurate (people don't differentiate cell value and cell averaged value). That's why QUICK does not show third order accuracy in FVM. However, if you evaluate QUICK in FDM, it will show its third order accuracy

I agree, while avaluating the order of accuracy people evaluate the discretization error but in a FVM, due to the integral formulation, one always computes the local averaged value in the cell. Therefore, you can use a flux reconstruction of arbitrary high order but the resulting solution is a second order approximation of the pointwise function. That appears particulary if one analyses also the QUICKES version where an explicit pointwise reconstruction from averaged value (a sort of deconvolution) is introduced to get third order accuracy in time and space.

[Avr.Tomer]

I wrote this post about numerical schemes and unwinding quite a while ago. It also relates to another important and sometimes unregarded issue of non-physical oscillation of the numerical solution from one grid point to the next which does not arise simply from nonlinearities of Navier–Stokes equations (NSE), but rather its source is the discretized linear equation.
I think it captures communicates the essence regarding upwinding very well:

https://cfdisraelblog.wordpress.com/...nolds-problem/