I find the argument that upwind difference can dissipate high wave number modes is questionable. If we use Fourier analysis to solve the semi-discretized convection-diffusion equation, we can find that the viscous term produce physical dissipation. If the convective term is discretized by upwind difference, additional numerical dissipation is produced. However, at high wave number, the physical dissipation generated by the discretisation of viscous term is always smaller than its genuine value. The smaller part can only be offset by the numerical dissipation of upwind difference. Hence, I think upwind difference cannot produce additional numerical dissipation for high wave number mode. What do you think about that?

Maybe you should try to use it...

So, according to my analysis, upwind difference should be used in convective term. It will not do any harm?

I'm sorry...it was sarcasm. What i meant is that, if you have ever used it, you know how harmful the upwind scheme is.

I assume you are talking about the first order scheme...for higher order schemes you will need some kind of bounding and it can be classified as a different kind of scheme.

However, asymmetrical schemes will always introduce some kind of dissipation because the modified wave number will be complex

I mean the 5th or higher order upwind schemes for the convective term.

By the way, I still not quite understand the reason that if the modified wave number has an imaginary part, then the schemes are dissipative.

Is this property independent of the PDE? Because most the books I read analyze the upwind scheme with linear wave equation. I can understand their explanations. However, I doubt whether the dissipative property of upwind difference still hold in nonlinear NS equations.

The only explanation of this property, independent of the PDE I read is in the paper of T.K Sengupta (1999). He explain it as the upwind scheme change the wave number from real k to complex k'. Then the part exp[Imag(k')*x]contribute to the change of amplitude. I find this explanation is Far-fetched. What's your opinion?

Thank you so much.

All of the analyses are based on linear equations, so the results are said to hold loosely when we go to the non-linear equations. The odd-order schemes typically pull in odd order derivatives when you back out the modified PDE, and it is these terms that modify the wave numbers from being real to having imaginary terms (at least if I remember correctly). The property is not strictly independent of the PDE, but since you are going to apply it to the first-order spatial derivative the behavior should be similar to what you see with the model equation.

Thank you so much for your answers.

One more question I need to ask. When we solve convection-diffusion equation, the dissipation generated by viscous term is always smaller than the genuine value at high wave number, because of the limitation of finite difference.

So, should I choose the upwind scheme that can generate the total dissipation (from viscous and convective term discretisation) equal to the genuine value (the actual physical dissipation that should be)?

Thank you

Upwinding is (as far as I am aware) never chosen with regard to the viscous terms. It is used to provide inherent stability to the convective term. The problem (as you are finding out) is that upwinding is much more dissipative than central differencing, and is thus not an automatic cure - running first-order upwind everywhere can be disastrous if you want to maintain sharp discontinuities. It may be possible to invoke some form of artificial damping (which is what upwinding provides intrinsically) to adjust the physical damping, but I would be uneasy trying to add damping via the convective term to make up for a limitation in the viscous term discretization. I would prefer to look for a fix in the viscous approximation.

With that said, I am not aware of any special care that is taken with regards to the viscous term - in most general purpose codes, second-order central differencing is used for the viscous terms.

I see. Thank you so much.