Are you trying to use Adam-Bashforth scheme for time-integration of the convection terms.

Atleast some in the LES community who use central-differencing for convection use 3rd order RK scheme for integration. I do not remember off hand what the stability characteristics of this scheme are but I think the CFL should be less than sqrt(3). Its stability also does not depend of the viscous terms the methods which use Adam-Bashforth scheme for convection and implicit/trapezoidal scheme for viscous terms.

Adrin,

I think you just identified on a potential source of confusion. The operator O(.) is indeed used to represent the order of magnitude of a quantity. But it is also used to indicate the way in which quantities scale. So O(h^2) doesn't necessarily mean the quantity is of order magnitude h^2. It can also mean that the quantity in question scales as h^2 (and the order of magnitude of the whole term might be unknown).

You get better accuracy with a 4th order scheme than with the 2nd order or 3rd order upwind biased schemes. However, if the grid is highly stretched or skewed (i.e., if the geometry is very complex), I have sometimes found that 5th (or even 3rd) order upwind schemes produce more accurate solution than 2nd or 4th order central schemes. Also, the central schemes are highly susceptible to boundary errors, especially from the outflow boundary.

As a general rule, formal order has little to do with the actual accuracy of the solutions. However, central schemes are preferred in LES since they have minimal artificial dissipation. 4th order scheme is preferrable since the difference between a fourth and second order approximations of the convective term can (sometimes) be as large as the subgrid stress contribution (computed from Smagorinsky or some other algebraic model). This means that the model contribution is comparable to the trunction error in the 2nd order scheme which means that the truncation error needs reduced further. One thing to realize is that the recommendations about schemes, time integration methods, grid layouts (staggered or non-staggered or semi-staggered) are made from simulation of incompressible flows in simple geometries. Sometimes, there isn't even a geometry, people simply use isotropic turbulence LES. Hence, everything needs to understood in context. For example, much is made of the kinetic energy conserving scheme proposed recently. However, you can not extend it to variable density flows. In fact, most central schemes create problems in the presence of strong density gradients (in combustion problems).

After being in research for so long, I am sure you will agree with me that if you write a paper on LES and say that you have used a 4th order central scheme, there will be no questions asked. If it happens to be a energy conserving scheme, even better. If you use a 2nd order scheme or an upwind-biased scheme, you could face problems from reviewers. You have the burden of explaining anything that is considered to be the norm in the LES community.

On the issue of backscatter modeling in LES, I have not seen proof from anyone that they are doing it right. In fact, some spectrum based eddy viscosity models do account for some local backscatter around the cut-off (which is what reduces the cusp in the spectral eddy viscosity at the cut-off) which incidentally is the most significant portion of backscatter. As far as non-local backscatter is concerned, I do think anyone is doing it and I have a feeling that it can not be done but that is separate discussion altogether.

That is what I was trying to point out. I might have gone a little extreme with my examples (but precisely to make a point).

I agree with the points Kalyan has made (so well) in response to my earlier postings.

The point is: First let's not confuse "scaling" and order of magnitude analysis because we end up making unfounded conclusions and sometimes outrageous claims (as seen in many journal papers). Second, having said the above, CFD is still an art as much as it is applied math and science. That is, while formally one may be able to show that one method is more accurate than another, "on the ground" the situation may be different. This is of course possible because the "formal" analysis was based on some assumptions that might not have been correct or valid for all ranges.

The example that Kalyan brought up with the 2nd- 3rd- and 4th- order methods and how a 3rd-order method _can_ in reality produce more accurate results than a 4th-order method basically is a verification of what I just said above.

I have to add, in order to avoid confusion, under similar conditions (that is using exactly the same discretization strategy) higher order schemes _are_ more accurate than their lower-order counterparts (and they also converge faster). There are of course exceptions to this rule - boundary or areas with sharp gradients may force one to use lower-order methods (or much more sophisticated higher-order methods)

I hope I haven't opened up another can of worms )

Adrin Gharakhani

(1). I think, most of the time, people without hands-on experience is making comments about an interesting problem. That's perfectly all right in the forum. (2). Your question about the method used by the researchers at the CTR at Stanford, is a valid one. Since it is a well-known university, it is important for the researchers at CTR to give us some insights of their papers or work. (3). The use of the second-order scheme for convection terms does not always lead to divergence. In my formulation it will. But I remembered long time ago, a professor from University of Cincinnati claimed that it depends on the solution procedure also. (4). So, we will hold the researchers at CTR responsible for this puzzle. We hope that they are reading this message, and have the courage to answer the question here.

(1). I have to say that, in the Taylor's series expansion, it is based on the fixed local mesh or cell size. (2). The higher-order terms in the expansion are "analytical" and are defined locally. (3). That's about all in terms of the order-of-magnitude analysis. (4). In the actual practice, higher-order terms or schemes must occupy more mesh points and cells. As a result of this, you are no longer talking about the same local cell. (5). This can be seen easily from the boundary points, where you will run into difficulty in approximating the higher-order schemes.