[CIG-MC] SUPG method in CitcomS

[Jed Brown]

On Fri, Apr 6, 2012 at 18:47, Matthew Knepley <knepley at mcs.anl.gov> wrote:

> Following on the recent question on the list about max temperatures in
>> CitcomS.
>> I wanted to ask about the SUPG solver for the energy equation.
>>
>> My basic questions are:
>>
>> - Do you always expect to have some "small" temperature oscillation from
>> the SUPG method?
>> - If so, in your opinion when is it reasonable to use the Lenardic filter
>> to prevent these small
>> oscillations from growing?
>> - If not, what strategy would you use to eliminate the temperature
>> overshoots when decreasing
>> the time-step doesn't eliminate them?
>>
>> For example, the slab benchmark in the manual gets temperature overshoots
>> if you run that
>> forward in time. Decreasing the time-step helps, but doesn't eliminate
>> them.
>>
>> I've attempted to read some recent papers related to SUPG (after reading
>> the earlier Brooks
>> and Hughes papers), and I'm finding it difficult to understand the method
>> well enough to know
>> what its limitations are in non-steady-state problems (with or without
>> strong flow gradients).
>>
>
> Amazingly, the first Google hit looks good:
> http://rzbl04.biblio.etc.tu-bs.de:8080/docportal/servlets/MCRFileNodeServlet/DocPortal_derivate_00001549/Document.pdf
>
> SUPG just introduces some diffusion in the streamline direction to prevent
> oscillation. Its a little
> goofy because you have the tune the amount of diffusion carefully. It does
> not guarantee monotonicity.
> The right thing to do is implement a monotonic method like TVD or WENO.
> There are now easy, open
> source implementations in http://numerics.kaust.edu.sa/pyclaw/.
>

I agree that a non-oscillatory method is a better choice. SUPG is only
first order accurate for transport-dominated flows and it's still
oscillatory (though more accurate than other common first order schemes).
Godunov's theorem says that to have a robustly non-oscillatory method that
is greater than first order accurate, you have to use a nonlinear spatial
discretization. The TVD and WENO families of finite volume methods, as well
as limited discontinuous Galerkin methods embrace this. For structured
grids, WENO tends to be the most robust and easy method to implement. For
unstructured grids, DG is a good choice.

Discontinuity capturing methods for continuous finite element methods tend
to be less robust than the finite volume and DG counterparts. The better
choices typically use an indicator based on the residual of an entropy
equation. The performance becomes sensitive to the choice of entropy
function and they still don't generally guarantee non-oscillatory solutions.


> You can use a low-pass filter (Lenardic)
> to get rid of the oscillations, but the implications for accuracy are not
> good and forget about conserving
> anything.
>

The Lenardic filter is actually globally conservative (but not locally) and
is intended to sharpen interfaces. A low-pass filter would tend to prevent
oscillations by making the field smoother. My understanding is that the
Lenardic filter is intended for material interfaces, not for thermal
diffusion. Is it really used for thermodynamics?

I assume you are familiar with this study showing how the filter affects
tracer dynamics?

http://jupiter.ethz.ch/~pjt/papers/Tackley2003GC_TracerRatioMethod.pdf


>
> I have said the simplest thing. Jed knows more about this.
>
>     Matt
>
> From reading of one paper (Bochev et al., Stability of the SUPG FEM for
>> transient advection-
>> diffusion problems, in Comput. Methods Appl. Mech Eng, v. 193, 2004, p
>> 2301-2323),
>> they point out that:
>> "Regarding the small localized oscillations in SUPG solutions we recall
>> that SUPG is not monotonicity
>> preserving, and that such oscillations can be expected in the vicinity of
>> discontinuities and internal layers.
>> Therefore, their presence cannot serve as an indication of a
>> destabilization. Moreover, as the data in
>> Tables 1–3 show, smaller time steps do not lead to an increase in the H1
>> seminorm of the solutions,
>>  i.e., these oscillations remain bounded for small time steps. An
>> application of a discontinuity capturing
>> operator [16] is recommended for a further suppression of these
>> oscillations.
>>
>
My opinion is that these guys are using analysis tools that aren't really
appropriate. It's interesting math, but tends not to acknowledge Godunov's
theorem and, in my opinion, gives much less useful results than the finite
volume/DG community (see Chi-Wang Shu's various review papers for that
perspective).

Here is a recent review comparing transport schemes. It's still written
from a FEM perspective, but branches out in a good way.

http://www.wias-berlin.de/people/john/ELECTRONIC_PAPERS/JN12.JCP.pdf

I'll also mention that there are some newer exotic high-order methods such
as "spectral difference" that have some nice properties, but still aren't
very mature.
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