[CIG-CS] CIG-CS Digest, Vol 12, Issue 4
mspieg
mspieg at ldeo.columbia.edu
Tue May 31 19:50:04 PDT 2011
Hi, lots of messages here...I'll just try to respond to the ones
with respect to methods I've used.
>
> ------------------------------
>
> Message: 2
> Date: Tue, 31 May 2011 13:10:10 -0700 (PDT)
> From: Walter Landry <walter at geodynamics.org>
> Subject: Re: [CIG-CS] Particle problems and Semi-Lagrangian Schemes
> for Gamr
> To: cig-cs at geodynamics.org
> Message-ID:
> <20110531.131010.1281843293629835592.walter at geodynamics.org>
> Content-Type: Text/Plain; charset=iso-2022-jp-3
>
> Back to the list.
>
> mspieg <mspieg at ldeo.columbia.edu> wrote:
>> Hi Walter,
>> Mike is quite right, I've been using Semi-Lagrangian methods for
>> ages in
>> both finite volume, and more recently finite element
>> calculations. They have
>> a lot of nice properties, but can be computationally expensive in
>> elements
>> (depending on how they're implemented). However, because they advect
>> continuous fields, they will scale linearly with your degrees of
>> freedom (or
>> possibly quadrature points) (and they don't have CFL stability
>> issues).
>
> This is very interesting. What do you mean that they are expensive in
> elements? I would expect the cost of advection would be dwarfed by
> the cost of solving Stokes.
>
It depends on how you implement it. I believe that the
mathematically correct way to do this in elements is to solve the
weak form of the semilagrangian advection equation...which becomes a
projection problem
for example pure advection of a scalar field f can be written using
the material derivative as
Df/Dt = 0
or along a characteristic, the strong form is
f(x,t+dt) = f(x*(t),t)
where x*(t) is the take-off point for the characteristic that
intersects point x at time t+dt
the weak form, given a test function v is
\integral v*f dx = \integral v*f*
which is a projection problem. What is being projected is the
distorted advected field f* onto the function space. To evaluate the
RHS, you want the value of f* at the quadrature points, which is
simply the value of f at the take-off point at time t for the
characteristic that intersects the quadrature point at time t+dt
(i.e. for elements you calculate take-off points for quadrature
points, not for degrees of freedom like in finite volumes or finite
differences). Depending on your quadrature rules, you can have more
quadrature points than degrees of freedom, but it is a fixed multiple
of your number of elements (and many fewer than the number of
particles used in pic schemes)
On an unstructured mesh, finding the take-off points can be expensive
as is evaluating all the basis functions at arbitrary take off
points. For a structured or semi-structured (e.g. octree) mesh,
finding the take off points is much easier. Still the results are
quite good if you can amortize the cost by taking large steps and it
preserves the symmetry of the underlying discrete matrices.
> Also, I found Katz et. al. [1], which includes a mid-ocean ridge
> model. That uses finite volumes. Do you have any results for a
> finite-element semi-lagrangian scheme I can look at?
>
yes. I have a small Semi-Lagrangian class built on Dolfin Functions
with some test programs in them. I think they're current up to
release 0.9.9. of dolfin. I have them as a mercurial repository and
can post them to CIG.
>> I actually have a demonstration T-Dependent viscosity Rayleigh-
>> Benard problem
>> with semi-lagrangian advection implemented in hybrid FEniCS/PETSc
>> codes if
>> you're interested.
>
I use the semi-lagrangian method in this Rayleigh benard code even
for parabolic equations. The SLCN method for finite differences is
discussed in our G-cubed paper
@Article{Spiegelman2006,
author = {Marc Spiegelman and Richard Katz},
title = {A Semi-Lagrangian Crank-Nicolson Algorithm for the
Numerical Solution of Advection-Diffusion Problems},
journal = G3,
year = 2006,
volume = 7,
pages = {Q04014},
note = {doi:10.1029/2005GC001073}
}
> Cheers,
> Walter Landry
> walter at geodynamics.org
>
>
> [1] Numerical simulation of geodynamic processes with the Portable
> Extensible Toolkit for Scientific Computation
> R.F. Katz, M.G. Knepley, B. Smith, M. Spiegelman, E.T. Coon,
> Physics of the Earth and Planetary Interiors 163 (2007) 52?#|68
>
>
> ------------------------------
>
> Message: 3
> Date: Tue, 31 May 2011 15:13:04 -0500
> From: Matthew Knepley <knepley at mcs.anl.gov>
> Subject: Re: [CIG-CS] Particle problems and Semi-Lagrangian Schemes
> for Gamr
> To: Walter Landry <walter at geodynamics.org>
> Cc: cig-cs at geodynamics.org
> Message-ID: <BANLkTimuXmoG9T1trU7UFw5yYyQ1CrU-8g at mail.gmail.com>
> Content-Type: text/plain; charset="utf-8"
>
> 2011/5/31 Walter Landry <walter at geodynamics.org>
>
>> Back to the list.
>>
>> mspieg <mspieg at ldeo.columbia.edu> wrote:
>>> Hi Walter,
>>> Mike is quite right, I've been using Semi-Lagrangian methods
>>> for ages
>> in
>>> both finite volume, and more recently finite element
>>> calculations. They
>> have
>>> a lot of nice properties, but can be computationally expensive in
>> elements
>>> (depending on how they're implemented). However, because they
>>> advect
>>> continuous fields, they will scale linearly with your degrees of
>>> freedom
>> (or
>>> possibly quadrature points) (and they don't have CFL stability
>>> issues).
>>
>> This is very interesting. What do you mean that they are
>> expensive in
>> elements? I would expect the cost of advection would be dwarfed by
>> the cost of solving Stokes.
>>
>
> I suspect you are correct for the exceedingly simple geometry of
> GALE. If
> GAMR
> is just octrees, you are also right AND there is code for this in
> PETSc
> already.
>
> Matt
>
Yes...Matt and Rich wrote a parallel SemiLagrangian code on DA's. I
think it's the Characteristics class
>
>> Also, I found Katz et. al. [1], which includes a mid-ocean ridge
>> model. That uses finite volumes. Do you have any results for a
>> finite-element semi-lagrangian scheme I can look at?
>>
>>> I actually have a demonstration T-Dependent viscosity Rayleigh-
>>> Benard
>> problem
>>> with semi-lagrangian advection implemented in hybrid FEniCS/PETSc
>>> codes
>> if
>>> you're interested.
>>
>> Cheers,
>> Walter Landry
>> walter at geodynamics.org
>>
>>
>> [1] Numerical simulation of geodynamic processes with the Portable
>> Extensible Toolkit for Scientific Computation
>> R.F. Katz, M.G. Knepley, B. Smith, M. Spiegelman, E.T. Coon,
>> Physics of the Earth and Planetary Interiors 163 (2007) 52?#|68
>>
>> _______________________________________________
>> CIG-CS mailing list
>> CIG-CS at geodynamics.org
>> http://geodynamics.org/cgi-bin/mailman/listinfo/cig-cs
>>
>>
>
----------------------------------------------------
Marc Spiegelman
Lamont-Doherty Earth Observatory
Dept. of Applied Physics/Applied Math
Columbia University
http://www.ldeo.columbia.edu/~mspieg
tel: 845 704 2323 (SkypeIn)
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