Hello Clemens,
Hum, excellent remarks and observations. The option 2 of SUPG adds the amount of upwinding that just gives the stability, while option 1 gives a full upwinding, so option 2 is more accurate than option 1 if the Courant number is less than 1. For Courant numbers larger than 1... well I am not sure of what happens... but normally option 2 should be more stable, but at the cost of an extra diffusion, this is why it is not recommended in the user manual in such situations, though it would be more stable (I admit it is somewhat confusing..).
Your remarks on wiggles are perfectly correct. When we have a linear interpolation for both velocity and depth, wiggles may occur but not always, this is the inf-sup condition theory (actually nobody really proved that Saint-Venant equations are prone to inf-sup oscillations, but a fact is that sometimes we see them).
We favour another possibility to suppress wiggles :
No upwinding on SUPG and :
COMPATIBILITY OF FREE SURFACE GRADIENT : 0.9
this keyword may be between 0 and 1
This new technique consists of considering that the gradient of the free surface is a function which is constant per element (this is normal as the free surface is piece-wise linear, but finite elements do not do that because the free surface gradient contributes to velocities which are linear). When finite elements average the free surface gradients to get a linear function, they do not see the wiggles, whereas when you keep the free surface gradients as piece-wise constant functions, you see the wiggles and it creates the velocities that will smooth them.
COMPATIBILITY OF FREE SURFACE GRADIENT : 1 is the default value that does not see the wiggles (hence the need of SUPG upwinding). With values less than 1 the velocities are considered for some time in the algorithm as a sum of a linear and of a piece-wise constant function.
this new option has been tried successfully in cases where wiggles did occur.
I hope this helps, you pointed out a fundamental problem of Shallow water and Navier-Stokes equations,
With best regards,
Jean-Michel Hervouet