Thank you both, Jonathan and Daniel, for the most remarkably detailed assistance and insights!! Your replies have clarified numerous confusions someone who's a novice in numerical analysis would probably have. I ran through the routine again with the fixes made and the video is attached. As expected, only the CentralDifference method fails under a high Peclet value. By the way, is there a way for FiPy to calculate and display that? According to the FiPy doc http://www.ctcms.nist.gov/fipy/documentation/numerical/discret.html#linear-equations, the advection strength requires the dot product of the advection component and the normal vector at each face, yet the latter value is not obvious to me as an end user not knowing the topology of the solution domain.
Cheers, Yun On Wed, May 2, 2012 at 8:57 AM, Jonathan Guyer <[email protected]> wrote: > > On May 2, 2012, at 11:35 AM, Daniel Wheeler wrote: > > > On Tue, May 1, 2012 at 8:38 PM, Yun Tao <[email protected]> wrote: > > >> 2. Out of the other six, the Central Different Convection Term appears > to > >> partially fail with the following message: > >>> > >>> MaximumIterationWarning: Iterations: 1001. Relative error: 1.61715 > > > > Try switching the solver to LU "solver=LinearLUSolver()" in the solve > > arguments. You're at a Peclet number above 2 so Central Difference > > will break down. > > > >> 3. Explicit Upwind produces something vastly different from the rest > while > >> the regular Upwind generates something much closer. > > > > As I said, explicit is nonsense without a sensible time step. > > > >> These results make me worry about the package's numerical accuracy, > >> especially as I'm more interested in the transient, rather than > >> steady-state, solutions. > > > > They are all known issues for those particular first order schemes. > > First order schemes are good because they are completely implicit and > > so you can use large time steps when spatial accuracy is less of a > > consideration. Often when you just want to see the trends of a problem > > and can't solve with small time steps and when you > > are not solving interface tracking problems but smooth fields. For > > high accuracy you are restricted by the cfl (at least fot finite > > volume) > > Just to clarify what Daniel is saying, CentralDifference assumes that the > problem is diffusion dominated (Peclet is small), whereas Upwind assumes > that the problem is convection dominated (Peclet is large). The parameters > you have are neither large nor small so neither scheme does very well. > Exponential, Hybrid, and PowerLaw all interpolate between CentralDifference > and Upwind depending on the Peclet number, so all do OK for this problem. > Explicit and VanLeer just don't make sense for a steady-state problem. > > >> On a more trivial note, as seen in adv-dif_fipyQ.py, the placement of > the > >> solvers setup relative to the viewers setup (# PRE-solver vs. # > >> POST-solver) changes the presentation of the graphical outputs, > although the > >> numerical solutions remain the same. Is there then an optimal way to > >> illustrate the results? > > Either call viewer.plot() after creating the viewer to provoke > auto-scaling (which is what's happening in your POST case) or explicitly > set datamin and datamax when you create the viewers. > > _______________________________________________ > fipy mailing list > [email protected] > http://www.ctcms.nist.gov/fipy > [ NIST internal ONLY: https://email.nist.gov/mailman/listinfo/fipy ] > -- Graduate Group of Ecology Doctoral Candidate Department of Environmental Science and Policy Center for Population Biology University of California, Davis
7convections_movie.py
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