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Hello,

I have some questions regarding the SUPG option. In most cases I use for Type of advection 1;5;1;1. So for the conservative scheme + SUPG (5) I have to specify the SUPG option. When I choose NO Upwinding for H, SUPG OPTION 1;0, in some cases I have an unstable scheme, for example wiggles at the inflow boundary. In other cases this configuration works well (also if Courant number > 0.6). Generally No Upwinding would give the most accurate solution?
How can I determine apriori the appropriate SUPG option? Or should I always start with SUPG option 0, and in the case of an unstable calculation proceed to SUPG option 1 or 2?
I'm a little bit confused about the different considerations given in the User Manual and in the Reference Manual:
User Manual:
"In principle, option 2 is more accurate when the Courant number is less than 1 but
must not be used for large Courant numbers."
Reference Manual:
"The option 2 which was deduced from Fourier
analysis, gives the minimum diffusion to ensure the greatest stability of the scheme (voir [11]).
This option is the most stable when the Courant number exceeds 1."

I would be happy for some explanations and user experiences.

Best regards,
Clemens

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

Dear Jean-Michel,

thank you very much for the answer!
In my test case I increased the constant eddy viscosity, so that the Peclet number became very low Pe<<1. Increasing the velocity diffusivity generated strong wiggles at the inflow boundaries.
With the option FREE SURFACE GRADIENT COMPATIBILITY = 0.9 or SUPG = 1 for the depth I could overcome the instabilities at the inflows. As far as I understand, the SUPG option adds an artificial diffusivity to get more diffusion (->lower Peclet number).
Why does the increase of velocity diffusivity create these wiggles at the inflow boundaries?
Furthermore I don't understand why the SUPG stabilises the inflows although the Peclet number is very low?

Best regards,
Clemens

Hello,

Well, too much diffusivity is not always more stable, the wiggles could be due to the boundary terms. Think that when velocities are prescribed at the entrance, the momentum equation which includes diffusion is not solved there but only for the interior points, this may create small jumps in the solution. As for why SUPG manages to stabilise the solution with low Peclet number, it could be due to the semi-implicit treatment, while diffusion is fully implicit. Paradoxically implicit diffusion computes the diffusion terms with final values, i.e. values which have been smoothed by diffusion, so with lesser gradients, so in some sense implicit diffusion is less diffusive than explicit or semi-implicit diffusion. These are only hints, it would require a thorough Fourier analysis to be sure.

Regards,

Jean-Michel Hervouet