open TELEMAC-MASCARET The mathematically superior suite of solvers

Dear all,

sorry, I have a bit complicated matter.

A few years ago the non-conservative PSI advection scheme for velocities was the only really attractive one when (a) applying larger time steps in (b) the wave equation option for the free surface and (c) computing in parallel. However, in the institution I work for, it was noticed that if one is not willing do decrease the time step (ehm), it is enough to increase the number of internal iterations in the routine bief::vgfpsi above the hard-coded limit of 100 in order to pass through some parts of the simulations without an emergency stop in vgfpsi. (Yes it is true.) In our case this appears usually at the very beginning (river flows), when the usually awkward boundary conditions cause overshooting velocities in some places, especially when drying/wetting is involved.

Anyway, the number of internal iterations in vgfpsi is steered by a stability criterion being directly the Courant number Cr (computed in cflpsi). The number of iterations is simply the maximum value found in the whole computational domain rounded up to an integer. Well, having Cr>100 should force people to re-think their model setting, but even when at least the time step is reduced and BCs made less awkward, there is quite a lot of iterations in vgfpsi.

The reason for this is the fact that we apply for Telemac meshes with very variable space resolution and - note well - the same mesh for very various water levels. The fine elements tend to appear at the bathymetry slopes - just in order to get the volume correctly - where also the variable shoreline appears. Fine elements are sometimes always dry, sometimes overflown (with flood speeds!) causing the Courant numbers to vary strongly in the computational domain. This causes internal advection iterations also in places, where the stability criterion is well fulfilled, i.e. where these iterations are not necessary. I may imagine this increases the numerical diffusion of the scheme, although for most upwind schemes the less Cr, the more accurate the results are, so that dividing the time step might be not so bad as it seems.

My questions are:

(1) Are there any investigations about the accuracy of this PSI scheme as the function of the Courant number Cr?

(2) Are there any investigations how the scheme accuracy behaves with increasing number of internal iterations for a given constant time step?

(3) Is it theoretically and practically possible to iterate using the LOCAL value of the Courant number instead of the MAXIMUM value in the whole domain?

(The literature of the subject by numerous authors, most recommended by Roe, Deconinck, Hubbard, Struijs, etc. is too large - I ask only for the specific Telemac solutions.)

Best regards,
jaj

PS. Yes, I know there are other beautiful advection schemes one should try when applying larger time steps... ,^) jaj

Hello Jacek,

Glad to see you back on Telemac matters, I hope you are not starting a competition for the largest number of posts at the next user club, or the longest (joke!).

To answer your questions 1 and 2, nothing systematic has been done, it is difficult to do a Fourier analysis including iterations, and non linearity as there is in the PSI scheme. I would say that the quality is not decreasing as you increase the Courant number, as the scheme does its own sub-iterations, it may be rather insensitive to this. One thing is sure: for the same duration, more iterations will be more diffusive. It is the reason why the PSI scheme is better than the N scheme, just because it looks closer at the Courant number and finds a way to have larger time steps, by combining fluxes that will counterbalance each other.

Now question 3. If you look at the variants of distributive schemes which we have added recently for tidal flats(see release notes 6.0, now schemes 13 and 14 as variants of schemes 3 and 4). They actually use a local Courant number in the sense that if the fluxes can be transferred in one iteration (without negative depth) they will. The iterations of these new schemes are done only in places where there are difficulties, i.e. with large Courant numbers. By adding printouts in these new schemes, you can see the number of segments which are still involved in the iterations, and it rapidly decreases.

A possible progress would be to have on one hand implicit schemes, and on the other hand more accurate space discretisation, and as you mention, such schemes exist, some are being programmed (maybe my colleagues at HRW can comment on this), but everything must work in parallel, so it's an extra difficulty.

With best regards,

Jean-Michel Hervouet