Hello,

the title of your topic is not clear.... The Euler equations are always hyperbolic, when you discretize them by means of upwinding schemes you can see that the corresponding modified equations (the effective PDE that the scheme really satisfies) are not. They will present additional terms that, owing to their dissipative nature, change the mathematical character of the equations.

If the discretization is done without using dissipative schemes, you get always a modified equation with additional terms. In such case you can see dispersive-like terms.

What do you mean for "convergence", the reached steady state or the numerical convergence from time step to time step?

Hi Filippo,

Thanks for your answer. I mean the numerical convergence, and specifically in order to achieve numerical convergence why we need additional dissipative terms in Euler scheme when seen from mathematical perspective

Well, first let me say that is not necessary to introduce an artificial dissipation to get convergence... for hyperbolic equations many numerical formulations can be applied, I suggest to have a look to the LeVeque's book.

To address a mathematical point, remember that hyperbolic (non-linear) equations can produce a singularity in the solution even starting from a regular initial field and BC.s (as example consider the Burgers equation). Mathematically, the PDE (strong formulation) can no longer be used and a weak formulation is generally introduced. 

Numerically, you cannot describe the singularity onset on a finite-size grid and often high gradients can lead to spurious numerical oscillations into the solution (or even producing numerical instability). Using a weak formulation, you work with smoothed numerical solutions that are somehow the counterpart of working with the PDE where the numerical scheme adds artificial viscosity. That will dump the highest numerical wavenumbers into the numerical error.

Have you worked on some specific problem with no success in getting solution?