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 ]
