Many scientific journals require at least second order accurate discretization of the convective terms.

As I read it, from the methodology index, this is true for LUD, MARS, and naturally for QUICK. QUICK may result in solutions outside physical bounds, so I would prefer MARS (or LUD), but no reference is given to these 'non-classical' schemes, which restricts the 'journal-acceptability'.

Any suggestion on what scheme to choose? and what arguments to give in an article to justify the choice?

Thanks

DAN

(1). If you can obtain the mesh independent solution from your cfd calculations, then submit the results to a journal which does not require higher-order convection schemes. (2). For coarse mesh solutions, in some cases, higher-order schemes tend to give better solution relatively. But this is not always the case. The mesh independent solution is a must,because the solution is repeatable. (3). A non-repeatable higher-order scheme solution is not a good solution. (4). My suggestion is: try two schemes and obtain mesh independent solutions. Then validate it to see which solution is better.

(1) I do not agree. There are good reasons for the journals to require higher order schemes. See e.g.

B. P. Leonard "Comments on the Policy Statement on Numerical Accuracy". J. of Fluids Engineering, vol 115, 1993 pp 339--340.

One reason is that the 1. order upwind actually solves a 'different' problem than was intended, even for a grid-independent solution.

(2) The fact that in some (almost all) cases, higher-order schemes are superiour on coarse meshes is because the truncation error (for a given grid) is smaller for a higher order method. This also means that a grid-independent solution requires significantly more grid points for a 1. order method than for a higher order method!

(3) Naturally, the solution must be grid-independent and consistent.

(4) The 'best' solution for one setup is not necesserily the 'best' solution for other setups. I would rather know, beforehand, that I have used settings which (propably) ensure reasonable solutions.

Regards

DAN

I suggest you to see all refereces of B. P. Leonard because he's interested a bout this subject for long time.

I suggest you to see this references, you can find many ideas.

** P. Tamamidis, and D. N. Assanis, "Evaluation of Various High-order-accuracy Schemes with and without Flux Limiters", International Journal for Numerical Methods in Fluids, 16(10), 931-948, 1993.

** M. K. Patel, and N. C. Markatos, "An Evaluation of Eight Discretization Schemes for Two-dimensional Convection-Diffusion Equations", International Journal for Numerical Methods in Fluids, 6(3), 129-154, 1986.

** M. K. Patel, N. C. Markatos, and M. Cross, "A Critical Evaluation of Seven Discretization Schemes for Convection-Diffusion Equations", International Journal for Numerical Methods in Fluids, 5(3), 225-244, 1985.

** D. L. Roberts, and M. S. Selim, "Comparative Study of Six Explicit and Two Implicit Finite Difference Schemes for Solving One-dimensional Parabolic Partial Differential Equations", International Journal for Numerical Methods in Engineering, 20(5), 817-844, 1984.

** A. Rigal, and G. Aleix, "Stability Analysis of Some Finite Difference Schemes for the Navier-Stokes Equations", International Journal for Numerical Methods in Engineering, 12(9), 1399-1405, 1978.

** John C. Strikwerda, "High-order-accurate schemes for incompressible viscous flow", International Journal for Numerical Methods in Fluids, 24(7), 715-734, 1997

** Alexander G. Churbanov, Andrei N. Pavlov, and Peter N. Vabishchevich, "Operator-splitting methods for the incompressible Navier-Stokes equations on non-staggered grids. Part 1: First-order schemes", International Journal for Numerical Methods in Fluids, 21(8), 617-640, 1995

**Carlos M. Lemos, "Higher-order schemes for free surface flows with arbitrary configurations", International Journal for Numerical Methods in Fluids, 23(6), 545-566, 1996.

(1). Item-3, a good idea. But, just go back and review the papers published in the numerical schemes area, and count the number of papers which actually presented the mesh independent solutions. (2). I am using the upwind methods all the time,(including the commercial codes), it is similar to the velocity measurement, using Pitot probe, hot wire, or laser velocimeter. (3). If you are using a higher-order scheme, and the solution has wiggles in it for the particular problem you are trying to solve, then is it more accurate than the smooth upwind solution?

In general cases it's the upwind scheme gives good results but development in Taylor series of upwind scheme is only first order, so, if a higher-order scheme is used, and the solution for the specific problem has wiggles it's depended of how kind of scheme we use ; we can for exemple use a very simple and very recommended scheme like "Hybrid" order 1.

For the other schemes with and without Flux Limiters it's much difficult to talk a bout them. Try to see this book, you want to find many exemples in order to undrestand first how to make a comparaison between schemes

An Introduction to Computational Fluid Dynamics: The Finite Volume Method

By: Versteeg, H.K. and Malalasekera, W. Addison-Wesley, 1996

(1). You are talking about the schemes, while I am talking about the solutions. (2). Even with the upwind solution as the initial guess, it is still very difficult to get the solution to converge, when I switch to the higher-order scheme. (3). And if I use the higher-order scheme right from the begining, the solution does not converge unless I adjust the time steps or the relaxation factors many times in many steps.

One of the reasons why the first order scheme is easier to converge is that it is numerically dissipative, thus increasing the artifical (numerical) viscosity in the solution. The second order scheme tends to be dispersive, creating the oscillations that you see in the solution, especially in your pressure or Mach number plots, thus making it harder to converge. This has to do with the mathematics of the numerical solutions, and the previous discussion lists a good reference to explain this. I guess it depends on your problem to which scheme that you want to use...perhaps a hybrid scheme like someone listed before so that you don't lose too much accuracy but you introduce enough "stability" to converge your solution.

(1). Good suggestions. (2). Now, in order to get higher-order scheme solutions, we have to become an artist first. (If on the top of it, there is uncertainty in turbulence models, and lower order boundary conditions, the the whole thing is really a mess.)(3). I think, if one is interested in publishing papers in journals, it is a good idea to invent higher-order schemes.

In order to get higher-order scheme solutions, it's important to be an artist in Flux Limiters.

Dan, I've just started trying to use StarCD and I have some suggestions on getting references. (1)there is a reference for LUD: Wilkes, N.S. and Thompson, C.P. 1983 "An Evaluation of Higher Order Upwind Differencing for Elliptic Flow Problems" CSS 137, AERE, Harwell. (2)SFCD, Gamma differencing, and Blended Differencing are hybrid schemes discussed in Culbert Laney's book "Computational Gasdynamics". go to http://capella.colorado.edu/~laney.(3)MARS looks like one of the higher order reconstruction-evolution (in contrast w/ limited) schemes discussed also in Laney's book. (3) another way too look for refs is to search the journals for articles written by others using StarCD including employees of CD which may have the appropriate references. often they'll write a paper as a marketing tool to showcase new improvements. the people at fluent do this all the time I don't know about Cd though. Check CD's website which has links to StarCD user conferences (under news) where you may be able to obtain papers etc. (4) the most unlikely way is to ask the people at CD. they may or may not tell you. I've had other companies send me papers that they've published and others have papers online. CD may have another policy in this area but it doesn't hurt to try.

Dan,

in addition to what I wrote before you can go to http://www.adapco-online.com/frontpage.html in the "published works" section you'll see many articles written by StarCD users some of which look (by the journal to which they were submitted) to be useful. you may be able to find one that was submitted to the journal you're interested in.

Clifford

Thanks for the reply. I have discussed the subject with the CD people and got the reply on the MARS scheme from Mike Lewis that:

"We haven't published work on MARS because it is commercially sensitive. However, it is a TVD scheme, so you could use a reference to van leer and another to our methodology manual. (Other commercial codes have similar MUSCL type schemes)"

The van Leer reference is:

van Leer, B., Towards the ultimate conservative differencing scheme V: A second-order sequel to Godunov's method, J. Comp. Phys., 23 (1977), pp101-136

So, my solution is to write something like (where cite is a citation):

The SIMPLE algorithm was used with a second order MUSCL type scheme with a TVD limiter (\shortcite{vanLeer1977}) for the convective terms (denoted MARS in \shortcite{starcd1999}).

when i read up on the MARS scheme in the manual the description looked like some kind of ENO scheme so i guess the description is kinda misleading. As for "commercially sensitive" it's probably some warmed over Van Leer type limited scheme I guess.