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 ]
