# stabilize collocated grid via spline interpolated gradients ?

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November 1, 2013, 17:43
#2
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Filippo Maria Denaro
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Quote:

if you want to discretize in such a way, then you must discretize also Div(Grad p) as same way as you do for Div V . That will produce a very large stencil for the pressure

November 1, 2013, 18:57
#3
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Bernd
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Quote:
 Originally Posted by FMDenaro if you want to discretize in such a way, then you must discretize also Div(Grad p) as same way as you do for Div V . That will produce a very large stencil for the pressure

Yes I thought about this, but hoped to get away with just keeping the simple stencil for the pressure correction poisson equation. After all (at least in my projection based scheme) the pressure correction is only an approximation anyway and so I thought it might be sufficient to use my 27-point quadratic B-spline gradient discretization only for the gradients and only for the calculation of the right-hand-side of the pressure correction (and possibly for the pressure gradient but only for the result when adding it to the velocity after the poisson solve...
(somewhat similar to Chie-Row-Interpolation if I understood that correctly)

 November 2, 2013, 07:28 #4 New Member   Bernd Join Date: Jul 2012 Posts: 23 Rep Power: 7 After some more research I found a paper "A velocity-pressure Navier-Stokes solver using a B-spline collocation method" http://ctr.stanford.edu/ResBriefs99/botella.pdf dealing with the same idea. However, I'm afraid that the technical details of that paper are beyond my comprehension. In particular they mention that even with the B-spline discretization, the oscillation problem still occurs and they recommend a kind of "staggered B-spline basis" construction to overcome it. But this seems to be even more complicated than the regular staggered grid approch that I was trying to avoid wih the B-spline-approach .... So the error must be in my original assumption that it is the central difference gradient discretization alone that is responsible for the oscillation instability because with B-spline gradients, we use not only two but 27 neighboring grid cells for each gradient value and so I thought that this should not be prone to checkerboard oscillations any more ... but what do I know

 November 2, 2013, 10:24 #5 Senior Member   Filippo Maria Denaro Join Date: Jul 2010 Posts: 3,751 Rep Power: 41 but you want an exact or an approximate projection ?

November 2, 2013, 10:35
#6
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Bernd
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Quote:
 Originally Posted by FMDenaro but you want an exact or an approximate projection ?
an approximate (but stable, non-oscillating) projection would be sufficient.

The simulation is only for visualization/animation purposes, but the grid resolution is high (up to 1000x1000x1000) and unfortunately, at resolutions higher than about 200^3 instabilities/checkerboard/striping artifacts occur unless I use agressive artificial damping of the velocity field (which of course negates the effect of the high resolution :-/

 November 2, 2013, 10:44 #7 Senior Member   Filippo Maria Denaro Join Date: Jul 2010 Posts: 3,751 Rep Power: 41 are you sure this is not a symptom of numerical instability? maybe your time-step is to high for such a refined grid

November 3, 2013, 08:30
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Bernd
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Quote:
 Originally Posted by FMDenaro are you sure this is not a symptom of numerical instability? maybe your time-step is to high for such a refined grid
As I use an unconditionally stable semi-lagrangian advection scheme (cubic interpolation accuracy) it is normally not necessary to pay too much attention to the time step.
To the contrary: choosing too small a time step is even contraproductive with the semi-lagrangian schemes because it can lead to many repeated advection-interpolation steps 'within the same grid cell' and repeated interpolation = heavy blurring/numerical dissipation ...

So I know what I should really do is to switch the whole framework from
the 'collocated-grid/semi-lagrangian' to a 'staggered-grid/all-eulerian'
paradigm .... but this would mean to re-write almost all the code from scratch

However, on the other hand I really like the ability to take arbitrarily large time steps of the semi-lagrangian scheme (probably the only way to do a 1000^3 simulation on a normal workstation and the computational simplicity of the collocated grid ...

Concerning the co-located grid problem: maybe I schould take another look at the Rhie-Chow correction (but then again, wouldn't this mean to also change the discretization of the pressure gradient to a larger stencil in the poisson solver, which I try to avoid ?)

 November 3, 2013, 08:38 #9 Senior Member   Filippo Maria Denaro Join Date: Jul 2010 Posts: 3,751 Rep Power: 41 I worked on this problem some time ago, if you want check details here are you. However, the cost is a second elliptic solver... A 3D second-order accurate projection-based Finite Volume code on non-staggered, non-uniform structured grids with continuity preserving properties: application to buoyancy-driven flowsInternational Journal for Numerical Methods in Fluids Volume 52, Issue 4, 10 October 2006, Pages: 393–432, F. M. Denaro Article first published online : 10 FEB 2006, DOI: 10.1002/fld.1185 A non-diffusive, divergence-free, finite volume-based double projection method on non-staggered gridsInternational Journal for Numerical Methods in Fluids Volume 53, Issue 7, 10 March 2007, Pages: 1127–1172, A. Aprovitola and F. M. Denaro Article first published online : 27 SEP 2006, DOI: 10.1002/fld.1368

November 8, 2013, 03:49
#10
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Bernd
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Quote:
 Originally Posted by FMDenaro I worked on this problem some time ago, if you want check details here are you.
Thank you very much ! this turned out to be really helpful.

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