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 <[email protected]> wrote: > On Wed, Jul 20, 2016 at 1:30 PM, Raymond Smith <[email protected]> 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 > [email protected] > http://www.ctcms.nist.gov/fipy > [ NIST internal ONLY: https://email.nist.gov/mailman/listinfo/fipy ] >
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