Basic Numerics Question
 

Dear All,

I have some basic numerics question regarding 1st and 2nd order upwind scheme for the convective term for the 1d convection-diffusion equation:

u*rho*Phi' - Gamma* Phi'' = 0

When using the 1st order upwind scheme, one can see, that the first derivative in your differential equation is substituted by the first derivative minus dx/2 * second derivative.
Ok, so that means you will introduce some additional diffusion (namely u*rho*dx/2) into your equation, because diffusion has exact the same appearance.

When using the 2nd order upwind (or linear upwind scheme), the second derivative will cancel out in the error. That means there is no additional term that has exactly the same appearance as the diffusion.

So why do I read in every book that 2nd order upwind is also diffusive (but less then the 1st order)? Why is there additional diffusion at all?

The reasoning is based on general odd/even derivatives appearing in the local truncation error. A first derivative when discretized with upwind formulas produces a modified wavenumber that has both real and imaginary parts. These errors are associated to numerical dispersion and dissipation.
You can see the issue in the book of Peric and Ferziger.
If you want, more details are present here

http://onlinelibrary.wiley.com/doi/1...d.179/abstract

Hey Filippo,
I downloaded the paper. Thank you for the help.
To the Peric/Ferziger book: I have read some chapters several times, but unfortunately not the one you recommended ;)

Hi,
consider the section devoted to the analysis of the modified wavenumber. You will see that the local truncation error can be analysed also in the wavenumber space and you can see that any non-symmetric discretization of the first derivative produce an error that is never present in centred discretizations