Why numerical diffusion is related to the advection term
 

Hello All,

In many literature I found that the numerical diffusion is always related to the discretization of the advection term without any justification of this statement.
To the best of my knowledge the numerical diffusion is a result of the truncation error which exist when using the numerical method to discretize advection or diffusion term. I know the due to the nature of diffusion it allows the use of higher order scheme without facing the boundedness problem. But even higher order schemes still have some sort of truncation error.

Is there's somehing that I am not aware of?

Any discretization of a integral o differential operator produces a local truncation error, in this sense you are right, considering for example the momentum equation we have the time-derivative, the convection, the pressure and the diffusion terms that produce a global truncation error.
However, the key appears clear if you consider that only the diffusion term is of second order (for example in incompressible flows you have mu*Lap u). Therefore, when you discretize the other derivatives that are first order (time derivative and convective term), you have to check from the expression of the local truncation errors if appears some terms that has some coefficients that multiply a secondo order derivative. Such terms are the "numerical diffusion" as they mimics the diffusion already present in the equation but with a mgnitude (the coefficient) that is not physical but numerical.
Of course, the same physical diffusion when discretized has a truncation error but it appears with higher order derivatives, for example third or forth order derivatives (dispersion and dissipation, respectively).

One of the issues is that there are waves going in the positive and negative directions so the error term is a function of (dq(+)/dx - dq(-)/dx)/2. Physically this term is zero for a continuous system but numerically it is not. This, I believe, is the numerical diffusion term. If I understand the question correctly. As more terms are kept in the Taylor series, this gets closer to zero.

one can also see the local truncation error in the wavenumber space by using the modified wavenumber analysis...numerical diffusion is seen by presence of the imaginary part

Thanks Filippo,

Can you please elaborate on the physical meaning of the dispersion and dissipation because after some search I couldn't differentiate it from diffusion.

Best regards

By a physical point of view, the presence of a numerical diffusion is highlighted by the smoothing of sharp gradients. The presence of numerical dispersion is highlighted by the fact that an exact solution constituded by a packet of waves moving at a certain velocity is altered, producing different velocity of propagation for each wavelenght.

Thanks again Filippo, that helped a lot