Re: Accuracy of DiffusionTerm for non-uniform mesh
On Wed, Jul 20, 2016 at 5:45 PM, Raymond Smith wrote:
> That makes sense. So, here
> https://gist.github.com/raybsmith/b0b6ee7c90efdcc35d6a0658319f1a01
> I've changed it so that the error is calculated as
> sum( (\phi-phi*)^2 * mesh.cellVolumes )
> and depending on the value I choose for the ratio of dx_{i+1} / dx_i, I get
> convergence for exponential spacing ranging from 2nd order (at that ratio =
> 1) and decreasing order as that ratio decreases from unity.
Thanks for that. I'm not entirely sure, but the integral as written on
line 44,
https://gist.github.com/raybsmith/b0b6ee7c90efdcc35d6a0658319f1a01#file-fipy_accuracy-py-L44
may still only be first order accurate for non-uniform grids. I'll try
and investigate this though and see if it matters.
--
Daniel Wheeler
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Re: Accuracy of DiffusionTerm for non-uniform mesh
That makes sense. So, here
https://gist.github.com/raybsmith/b0b6ee7c90efdcc35d6a0658319f1a01
I've changed it so that the error is calculated as
sum( (\phi-phi*)^2 * mesh.cellVolumes )
and depending on the value I choose for the ratio of dx_{i+1} / dx_i, I get
convergence for exponential spacing ranging from 2nd order (at that ratio =
1) and decreasing order as that ratio decreases from unity.
On Wed, Jul 20, 2016 at 5:16 PM, Daniel Wheeler
wrote:
> I think you need to wait by volume, basically you're calculating "\int
> ( \phi - \phi^*)^2 dV" for the L2 norm, where \phi^* is the ideal
> solution and \phi is the calculated solution.
>
> You may also need to do a second order accurate integral in order to
> see second order convergence.
>
> On Wed, Jul 20, 2016 at 5:03 PM, Raymond Smith wrote:
> > Thanks.
> >
> > And no, I'm not sure about the normalization for grid spacing. I very
> well
> > could have calculated the error incorrectly. I just reported the root
> mean
> > square error of the points and didn't weight by volume or anything like
> > that. I can change that, but I'm not sure of the correct approach.
> >
> > On Wed, Jul 20, 2016 at 4:25 PM, Daniel Wheeler <
> daniel.wheel...@gmail.com>
> > wrote:
> >>
> >> On Wed, Jul 20, 2016 at 1:30 PM, Raymond Smith wrote:
> >> > Hi, FiPy.
> >> >
> >> > I was looking over the diffusion term documentation,
> >> >
> >> >
> http://www.ctcms.nist.gov/fipy/documentation/numerical/discret.html#diffusion-term
> >> > and I was wondering, do we lose second order spatial accuracy as soon
> as
> >> > we
> >> > introduce any non-uniform spacing (anywhere) into our mesh? I think
> the
> >> > equation right after (3) for the normal component of the flux is only
> >> > second
> >> > order if the face is half-way between cell centers. If this does lead
> to
> >> > loss of second order accuracy, is there a standard way to retain 2nd
> >> > order
> >> > accuracy for non-uniform meshes?
> >>
> >> This is a different issue than the non-orthogonality issue, my mistake
> >> in the previous reply.
> >>
> >> > I was playing around with this question here:
> >> > https://gist.github.com/raybsmith/e57f6f4739e24ff9c97039ad573a3621
> >> > with output attached, and I couldn't explain why I got the trends I
> saw.
> >> > The goal was to look at convergence -- using various meshes -- of a
> >> > simple
> >> > diffusion equation with a solution both analytical and non-trivial,
> so I
> >> > picked a case in which the transport coefficient varies with position
> >> > such
> >> > that the solution variable is an arcsinh(x). I used three different
> >> > styles
> >> > of mesh spacing:
> >> > * When I use a uniform mesh, I see second order convergence, as I'd
> >> > expect.
> >> > * When I use a non-uniform mesh with three segments and different dx
> in
> >> > each
> >> > segment, I still see 2nd order convergence. In my experience, even
> >> > having a
> >> > single mesh point with 1st order accuracy can drop the overall
> accuracy
> >> > of
> >> > the solution, but I'm not seeing that here.
> >> > * When I use a mesh with exponentially decreasing dx (dx_i = 0.96^i *
> >> > dx0),
> >> > I see 0.5-order convergence.
> >>
> >> That's strange. Are you sure that all the normalization for grid
> >> spacing is correct when calculation the norms in that last case?
> >>
> >> > I can't really explain either of the non-uniform mesh cases, and was
> >> > curious
> >> > if anyone here had some insight.
> >>
> >> I don't have any immediate insight, but certainly needs to addressed.
> >>
> >> --
> >> Daniel Wheeler
> >> ___
> >> fipy mailing list
> >> fipy@nist.gov
> >> http://www.ctcms.nist.gov/fipy
> >> [ NIST internal ONLY: https://email.nist.gov/mailman/listinfo/fipy ]
> >
> >
> >
> > ___
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> > http://www.ctcms.nist.gov/fipy
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> >
>
>
>
> --
> Daniel Wheeler
> ___
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>
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Re: Accuracy of DiffusionTerm for non-uniform mesh
I think you need to wait by volume, basically you're calculating "\int ( \phi - \phi^*)^2 dV" for the L2 norm, where \phi^* is the ideal solution and \phi is the calculated solution. You may also need to do a second order accurate integral in order to see second order convergence. On Wed, Jul 20, 2016 at 5:03 PM, Raymond Smith wrote: > Thanks. > > And no, I'm not sure about the normalization for grid spacing. I very well > could have calculated the error incorrectly. I just reported the root mean > square error of the points and didn't weight by volume or anything like > that. I can change that, but I'm not sure of the correct approach. > > On Wed, Jul 20, 2016 at 4:25 PM, Daniel Wheeler > wrote: >> >> On Wed, Jul 20, 2016 at 1:30 PM, Raymond Smith wrote: >> > Hi, FiPy. >> > >> > I was looking over the diffusion term documentation, >> > >> > http://www.ctcms.nist.gov/fipy/documentation/numerical/discret.html#diffusion-term >> > and I was wondering, do we lose second order spatial accuracy as soon as >> > we >> > introduce any non-uniform spacing (anywhere) into our mesh? I think the >> > equation right after (3) for the normal component of the flux is only >> > second >> > order if the face is half-way between cell centers. If this does lead to >> > loss of second order accuracy, is there a standard way to retain 2nd >> > order >> > accuracy for non-uniform meshes? >> >> This is a different issue than the non-orthogonality issue, my mistake >> in the previous reply. >> >> > I was playing around with this question here: >> > https://gist.github.com/raybsmith/e57f6f4739e24ff9c97039ad573a3621 >> > with output attached, and I couldn't explain why I got the trends I saw. >> > The goal was to look at convergence -- using various meshes -- of a >> > simple >> > diffusion equation with a solution both analytical and non-trivial, so I >> > picked a case in which the transport coefficient varies with position >> > such >> > that the solution variable is an arcsinh(x). I used three different >> > styles >> > of mesh spacing: >> > * When I use a uniform mesh, I see second order convergence, as I'd >> > expect. >> > * When I use a non-uniform mesh with three segments and different dx in >> > each >> > segment, I still see 2nd order convergence. In my experience, even >> > having a >> > single mesh point with 1st order accuracy can drop the overall accuracy >> > of >> > the solution, but I'm not seeing that here. >> > * When I use a mesh with exponentially decreasing dx (dx_i = 0.96^i * >> > dx0), >> > I see 0.5-order convergence. >> >> That's strange. Are you sure that all the normalization for grid >> spacing is correct when calculation the norms in that last case? >> >> > I can't really explain either of the non-uniform mesh cases, and was >> > curious >> > if anyone here had some insight. >> >> I don't have any immediate insight, but certainly needs to addressed. >> >> -- >> Daniel Wheeler >> ___ >> fipy mailing list >> fipy@nist.gov >> http://www.ctcms.nist.gov/fipy >> [ NIST internal ONLY: https://email.nist.gov/mailman/listinfo/fipy ] > > > > ___ > fipy mailing list > fipy@nist.gov > http://www.ctcms.nist.gov/fipy > [ NIST internal ONLY: https://email.nist.gov/mailman/listinfo/fipy ] > -- Daniel Wheeler ___ fipy mailing list fipy@nist.gov http://www.ctcms.nist.gov/fipy [ NIST internal ONLY: https://email.nist.gov/mailman/listinfo/fipy ]
Re: Accuracy of DiffusionTerm for non-uniform mesh
Thanks. And no, I'm not sure about the normalization for grid spacing. I very well could have calculated the error incorrectly. I just reported the root mean square error of the points and didn't weight by volume or anything like that. I can change that, but I'm not sure of the correct approach. On Wed, Jul 20, 2016 at 4:25 PM, Daniel Wheeler wrote: > On Wed, Jul 20, 2016 at 1:30 PM, Raymond Smith wrote: > > Hi, FiPy. > > > > I was looking over the diffusion term documentation, > > > http://www.ctcms.nist.gov/fipy/documentation/numerical/discret.html#diffusion-term > > and I was wondering, do we lose second order spatial accuracy as soon as > we > > introduce any non-uniform spacing (anywhere) into our mesh? I think the > > equation right after (3) for the normal component of the flux is only > second > > order if the face is half-way between cell centers. If this does lead to > > loss of second order accuracy, is there a standard way to retain 2nd > order > > accuracy for non-uniform meshes? > > This is a different issue than the non-orthogonality issue, my mistake > in the previous reply. > > > I was playing around with this question here: > > https://gist.github.com/raybsmith/e57f6f4739e24ff9c97039ad573a3621 > > with output attached, and I couldn't explain why I got the trends I saw. > > The goal was to look at convergence -- using various meshes -- of a > simple > > diffusion equation with a solution both analytical and non-trivial, so I > > picked a case in which the transport coefficient varies with position > such > > that the solution variable is an arcsinh(x). I used three different > styles > > of mesh spacing: > > * When I use a uniform mesh, I see second order convergence, as I'd > expect. > > * When I use a non-uniform mesh with three segments and different dx in > each > > segment, I still see 2nd order convergence. In my experience, even > having a > > single mesh point with 1st order accuracy can drop the overall accuracy > of > > the solution, but I'm not seeing that here. > > * When I use a mesh with exponentially decreasing dx (dx_i = 0.96^i * > dx0), > > I see 0.5-order convergence. > > That's strange. Are you sure that all the normalization for grid > spacing is correct when calculation the norms in that last case? > > > I can't really explain either of the non-uniform mesh cases, and was > curious > > if anyone here had some insight. > > I don't have any immediate insight, but certainly needs to addressed. > > -- > Daniel Wheeler > ___ > fipy mailing list > fipy@nist.gov > http://www.ctcms.nist.gov/fipy > [ NIST internal ONLY: https://email.nist.gov/mailman/listinfo/fipy ] > ___ fipy mailing list fipy@nist.gov http://www.ctcms.nist.gov/fipy [ NIST internal ONLY: https://email.nist.gov/mailman/listinfo/fipy ]
Re: Accuracy of DiffusionTerm for non-uniform mesh
On Wed, Jul 20, 2016 at 1:30 PM, Raymond Smith wrote: > Hi, FiPy. > > I was looking over the diffusion term documentation, > http://www.ctcms.nist.gov/fipy/documentation/numerical/discret.html#diffusion-term > and I was wondering, do we lose second order spatial accuracy as soon as we > introduce any non-uniform spacing (anywhere) into our mesh? I think the > equation right after (3) for the normal component of the flux is only second > order if the face is half-way between cell centers. If this does lead to > loss of second order accuracy, is there a standard way to retain 2nd order > accuracy for non-uniform meshes? This is a different issue than the non-orthogonality issue, my mistake in the previous reply. > I was playing around with this question here: > https://gist.github.com/raybsmith/e57f6f4739e24ff9c97039ad573a3621 > with output attached, and I couldn't explain why I got the trends I saw. > The goal was to look at convergence -- using various meshes -- of a simple > diffusion equation with a solution both analytical and non-trivial, so I > picked a case in which the transport coefficient varies with position such > that the solution variable is an arcsinh(x). I used three different styles > of mesh spacing: > * When I use a uniform mesh, I see second order convergence, as I'd expect. > * When I use a non-uniform mesh with three segments and different dx in each > segment, I still see 2nd order convergence. In my experience, even having a > single mesh point with 1st order accuracy can drop the overall accuracy of > the solution, but I'm not seeing that here. > * When I use a mesh with exponentially decreasing dx (dx_i = 0.96^i * dx0), > I see 0.5-order convergence. That's strange. Are you sure that all the normalization for grid spacing is correct when calculation the norms in that last case? > I can't really explain either of the non-uniform mesh cases, and was curious > if anyone here had some insight. I don't have any immediate insight, but certainly needs to addressed. -- Daniel Wheeler ___ fipy mailing list fipy@nist.gov http://www.ctcms.nist.gov/fipy [ NIST internal ONLY: https://email.nist.gov/mailman/listinfo/fipy ]
Re: Accuracy of DiffusionTerm for non-uniform mesh
Hi Raymond,
I was just corresponding with James Pringle off the list about the
same issue for a Delaunay triangulated mesh. Here is my explanation to
him.
I think this is the non-orthogonality issue. I should have realized
this much sooner. The only place we seem to reference this issue is in
www.ctcms.nist.gov/fipy/documentation/numerical/discret.html where it
is stated "this estimate relies on the orthogonality of the mesh, and
becomes increasingly inaccurate as the non-orthogonality increases.
Correction terms have been derived to improve this error but are not
currently included in FiPy [13]."
See this for example,
https://www.researchgate.net/publication/242349760_Finite_volume_method_for_the_solution_of_flow_on_distorted_meshes
In the test mesh that you show, some of the triangles are highly
non-orthogonal to each other. This results in errors. You can view the
quantity of non-orthogonality in the mesh using
nonorth_var = fipy.CellVariable(mesh=mesh, value=mesh._nonOrthogonality)
fipy.Viewer(nonorth_var).plot()
The following code uses triangles, but with a perfectly orthogonality
and, thus, gives perfect results.
import fipy as fp
nx = 50
dx = 1. / nx
mesh = fp.Tri2D(nx=nx, ny=nx, dx=dx, dy=dx)
psi = fp.CellVariable(mesh=mesh)
psi.constrain(1.0, where=mesh.facesLeft)
psi.faceGrad.constrain(1.0, where=mesh.facesRight)
eq = fp.DiffusionTerm().solve(psi)
fp.Viewer(psi - mesh.cellCenters[0]).plot()
nonorth_var = fp.CellVariable(mesh=mesh, value=mesh._nonOrthogonality)
fp.Viewer(nonorth_var).plot()
raw_input('stopped')
As far as your Delaunay triangulation is concerned, finding a way to
smooth out / reduce the non-orthogonality would be one way to improve
the accuracy.
The FiPy docs definitely need to be more explicit about this issue.
Also, actually addressing the orthogonality issue for diffusion terms
at least would be good.
Cheers,
Daniel
On Wed, Jul 20, 2016 at 1:30 PM, Raymond Smith wrote:
> Hi, FiPy.
>
> I was looking over the diffusion term documentation,
> http://www.ctcms.nist.gov/fipy/documentation/numerical/discret.html#diffusion-term
> and I was wondering, do we lose second order spatial accuracy as soon as we
> introduce any non-uniform spacing (anywhere) into our mesh? I think the
> equation right after (3) for the normal component of the flux is only second
> order if the face is half-way between cell centers. If this does lead to
> loss of second order accuracy, is there a standard way to retain 2nd order
> accuracy for non-uniform meshes?
>
> I was playing around with this question here:
> https://gist.github.com/raybsmith/e57f6f4739e24ff9c97039ad573a3621
> with output attached, and I couldn't explain why I got the trends I saw.
> The goal was to look at convergence -- using various meshes -- of a simple
> diffusion equation with a solution both analytical and non-trivial, so I
> picked a case in which the transport coefficient varies with position such
> that the solution variable is an arcsinh(x). I used three different styles
> of mesh spacing:
> * When I use a uniform mesh, I see second order convergence, as I'd expect.
> * When I use a non-uniform mesh with three segments and different dx in each
> segment, I still see 2nd order convergence. In my experience, even having a
> single mesh point with 1st order accuracy can drop the overall accuracy of
> the solution, but I'm not seeing that here.
> * When I use a mesh with exponentially decreasing dx (dx_i = 0.96^i * dx0),
> I see 0.5-order convergence.
>
> I can't really explain either of the non-uniform mesh cases, and was curious
> if anyone here had some insight.
>
> Thanks,
> Ray
>
> ___
> fipy mailing list
> fipy@nist.gov
> http://www.ctcms.nist.gov/fipy
> [ NIST internal ONLY: https://email.nist.gov/mailman/listinfo/fipy ]
>
--
Daniel Wheeler
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Accuracy of DiffusionTerm for non-uniform mesh
Hi, FiPy. I was looking over the diffusion term documentation, http://www.ctcms.nist.gov/fipy/documentation/numerical/discret.html#diffusion-term and I was wondering, do we lose second order spatial accuracy as soon as we introduce any non-uniform spacing (anywhere) into our mesh? I think the equation right after (3) for the normal component of the flux is only second order if the face is half-way between cell centers. If this does lead to loss of second order accuracy, is there a standard way to retain 2nd order accuracy for non-uniform meshes? I was playing around with this question here: https://gist.github.com/raybsmith/e57f6f4739e24ff9c97039ad573a3621 with output attached, and I couldn't explain why I got the trends I saw. The goal was to look at convergence -- using various meshes -- of a simple diffusion equation with a solution both analytical and non-trivial, so I picked a case in which the transport coefficient varies with position such that the solution variable is an arcsinh(x). I used three different styles of mesh spacing: * When I use a uniform mesh, I see second order convergence, as I'd expect. * When I use a non-uniform mesh with three segments and different dx in each segment, I still see 2nd order convergence. In my experience, even having a single mesh point with 1st order accuracy can drop the overall accuracy of the solution, but I'm not seeing that here. * When I use a mesh with exponentially decreasing dx (dx_i = 0.96^i * dx0), I see 0.5-order convergence. I can't really explain either of the non-uniform mesh cases, and was curious if anyone here had some insight. Thanks, Ray fipy_convergence.pdf Description: Adobe PDF document ___ fipy mailing list fipy@nist.gov http://www.ctcms.nist.gov/fipy [ NIST internal ONLY: https://email.nist.gov/mailman/listinfo/fipy ]
