Re: Pathology of discrete diffusion

[Mark P. Line]

Jan Kreft wrote:
> 
> David Sumpter wrote:
> >
> > 1. I am using a technique similar to Diffuse 2d for the diffusion of
> > heat and am slightly confused by one aspect. Consider a portion of
> > lattice with temperatures and first order diffusion:
> >
> >   0
> >  080
> >   0
> >
> > If the diffusion constant is 1.0 and evaporation rate is 1.0 then the
> > centre value will update to 0 while the outside values will update to
> > 2. i.e.
> >
> >   2
> >  202
> >   2
> >
> > This appears somewhat unnatural for heat equations. If the value 8 came
> > from some source, you would not expect the source to be colder than
> > the surroundings on the next time step........
> >
> > Am I right about this? Is there a theoretical explanation?


I don't think I saw the original discussion here, so maybe there's
something I'm missing. Fill me in if I'm not answering the question
you're asking.

Fourier's law says that the rate at which thermal energy flows between
two points (or in the discrete case, between two adjacent lattice cells)
is proportional (by a coefficient of thermal conductivity) to the
negative gradient of temperature between the points (or cells). [Real
diffusion involves both storage and flow, so you'll also have to deal
with the material's thermal _capacity_, eventually.]

For a large delta-t, I think you'll find that the gradient sometimes
reverses sign and that temperatures damped-oscillate into the steady
state. As you decrease delta-t, the model will equilibrate with fewer
and more highly damped oscillations. In continuous time, thermal energy
flows smoothly over the gradient until the steady state is reached
(other things being equal).

But even in continuous time, I wouldn't be surprised to find damped
oscillations if you're starting from a thermally very heterogeneous body
(hot or cold spots) and if the material has a high lambda (thermal
conductivity).


> > 2. Has anyone got a reference they can give me discussing heat
> > diffusion in terms of lattices? I'd be very grateful.

I haven't seen these for a while, but I think they have part of what
you're looking for:

Myrup, L.O. (1969) "A numerical model of the urban heat island",
_Journal of Applied Meteorology_ 8: 908-918.

Sellers, W.D. (1973) "A new global climate model", _Journal of Applied
Meteorology_ 12: 241-254.

Wierenga, P.J./de Wit, C.T. (1970) "Simulation of heat transfer in
soils", _Proc. Soil Sci. Am._ 34: 845-848.


But you can always go check the journal devoted to the topic:

  _Numerical Heat Transfer. Part A, Applications_
  _Numerical Heat Transfer. Part B, Fundamentals_

and its precursor (same name, without the split). You could browse
through all the article titles in the CARL database if you don't have
easy access to the journals themselves.


-- Mark

(Mark P. Line  --  Bellevue, Washington  --  <address@hidden>)




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