[petsc-users] Fwd: SNES examples question

[Jed Brown]

On Sat, May 14, 2011 at 20:32, Boyana Norris <norris at mcs.anl.gov> wrote:

> I'm mainly interested in the parameter limits -- I know how to specify
> them, but I don't see any explanation of valid values in the code. So for
> ex20, how large can beta get and still be meaningful?
>

As a model problem or as "real" radiation-diffusion? I've seen at least
"3.5" in papers, but higher may be possible. I put "3.5" in quotes because
that paper also had a flux limiter and an ionization model so the expression
was a bit different. As a model problem, I think you can take it arbitrarily
large as long as the boundary conditions are such that it is always bounded
below by a positive constant (and Newton doesn't send it off to a negative
temperature). It's worth considering the Newton linearization for this
problem. The flux is

- T^beta \grad T

so the linearization applied to an increment dT is

- T^beta grad dT - (beta T^{beta-1} grad T) dT

which looks like isotropic diffusion with diffusivity T^beta plus advection
with "velocity"

- beta T^{beta-1} grad T .

You can estimate the cell Peclet number as

Pe_h = h beta T^{beta-1} |grad T| / T^beta = h beta |grad T| / T .

In general, larger values of beta will produce thinner boundary layers at
the right edge of the domain where T is small and grad T is large. Due to
entropy considerations, heat still flows from the warm side to the cold
side, thus advection never dominates diffusion at a converged state (but it
could, depending on the initial guess, at some earlier point in the
iteration). If you are just interested in the linear systems, this is a
pretty indirect way to get variable-coefficient advection-diffusion.


>
> For ex29, are you saying that it's always generating a singular linear
> system, or only for low viscosity and resistivity?
>

Actually, it looks like that was just ILU and non-pivoting LU producing
singular preconditioners. With no preconditioner, KSP estimates reasonable
condition numbers.
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