Hi,

I am working on stability analysis on upwind scheme approximation of N-S equation based on staggered grid. Any one know how to deal with such problem?

Thanks for your information

Robert

Hi robert,

What do you mean by "such" problem? You want to know the performance of upwind schemes? How familiar are you with upwind schemes?(first order,TVD,MUSCL-Monotone Upstream-Centered Conservation Laws scheme). Are you dealing with incompressible or compressible flows? First order schemes are highly consistent and stable. Normally you can make your code run safely with first order(for any case) but it has its own disadvantages. So you go for second order and in that case you have to use limiters. Be specific about your question so that you get straight answer Regards gita

Hi, gita,

Thanks for your concern and suggestion for further discussion. I am using first order upwind scheme for imcompressible flow such as water. I encountered one problem if the viscosity is very low or very high (different application), the flow became unresonable or the results was not symetric during symetric problem. That is why I want to do stability analysis. May I know what is your advice?

Thanks

Best Rgds

Robert

Hi gita,

I am using SMAC method to solve the N-S equations, so the momentum equation is used to get the first try value for velocities (explicit), then solve the poisson equation for potential function to adjust the tilde velocities to final ones.

I hope this is clear.

robert

Hi, Robert, here is just a trivial point. You mentioned that your results for high Reynolds may be unsymmetric for some symetric problem. This may be reasonable, due to the so-called `elliptic instability', and has been verified with two-side driven cavity (JFM, v336, 1997 or T.C.F.D, v14, 2001). Anyway, the low-order upwinding scheme is quite stable.

Hi Robert,

As Paul said your results could be reasonable and you need to refer to some journal articles along these lines. As far first order upwind is concerned,there is no stringent restriction on the cell Peclet no and you can use large delta x. You have to look into the truncation error expression. In the TE of FDA, you will have artificial diffusion term sitting. If that doesn't represent the actual physical diffusion,then you will have problem. You can do von Neumann stability analysis to find the bounds for your convective-diffusion terms. Assume that it's 1D problem,then in the TE expression you will have term udeltax/2(u is your convective vel).This is the additional term sitting and acts as artificial diffusion. If this is much smaller than your actual viscosity(nu),then it's fine. When they are of same order,then you have to be careful. It may introduce unwanted wiggles in your solution. Try second order upwind and see whther you face the same problem. If you want learn more about limiters(which you need for seonc order upwinds)there is a nice SIAM journal paper by Sweby. I can get you the details if you're interested(I don't remember the year off handedly) cheers

Hi, gita

Thanks for your important information. I expect your detailed information about the journal paper by Sweby. As you said, there is artificial diffusion |u|dx, normally, the viscosity nu is less than the artificial diffusion in my study. the results is totally wrong or not? I am checking with the papers paul suggested.

Best wishes

robert

Hi Robert,

The paper on limiters is Sweby,P.K "High Resolution schemes using flux limiters for Hyperbolic conservation laws", SIAM J Num Anal. vol. 21 (1984). Regards gita