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 <daniel.wheel...@gmail.com> 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 <smit...@mit.edu> 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 <smit...@mit.edu> 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 ] >
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