<div class="gmail_quote">On Sat, May 14, 2011 at 20:32, Boyana Norris <span dir="ltr"><<a href="mailto:norris@mcs.anl.gov">norris@mcs.anl.gov</a>></span> wrote:<br><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex;">
<div>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?</div></blockquote>
<div><br></div><div>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</div>
<div><br></div><div>- T^beta \grad T</div><div><br></div><div>so the linearization applied to an increment dT is</div><div><br></div><div>- T^beta grad dT - (beta T^{beta-1} grad T) dT</div><div><br></div><div>which looks like isotropic diffusion with diffusivity T^beta plus advection with "velocity"</div>
<div><br></div><div>- beta T^{beta-1} grad T .</div><div><br></div><div>You can estimate the cell Peclet number as</div><div><br></div><div>Pe_h = h beta T^{beta-1} |grad T| / T^beta = h beta |grad T| / T .</div><div><br>
</div><div>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. </div>
<div> </div><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex;"><div><br></div><div>For ex29, are you saying that it's always generating a singular linear system, or only for low viscosity and resistivity?</div>
</blockquote></div><br><div>Actually, it looks like that was just ILU and non-pivoting LU producing singular preconditioners. With no preconditioner, KSP estimates reasonable condition numbers.</div>