[CIG-MC] SUPG method in CitcomS

Matthew Knepley knepley at mcs.anl.gov
Fri Apr 6 20:20:52 PDT 2012

On Fri, Apr 6, 2012 at 8:48 PM, Jed Brown <five9a2 at gmail.com> wrote:

> 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.

Can't you also do WENO reconstruction on an FEM solution?

> 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?

Okay, that was not how it was explained to me. Is it like a Laplacian filer
(which would sharpen edges)? I don't see how this would reduce oscillation,
although I guess it could result from the solve for the conservation


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.

What most experimenters take for granted before they begin their
experiments is infinitely more interesting than any results to which their
experiments lead.
-- Norbert Wiener
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