Front-fixing method for moving boundary problem

Hi Jonathan and Daniel,

Jonathan, although I'm starting a new thread here, I first want to thank 
you for your once again very complete answer on the use of operator 
variables to get access to additional functions. I'm not 100\% sure yet 
whether I'll need to use one of those. Nevertheless, I found your answer 
extremely useful, it helped me a lot to understand more of the mechanics 
of FiPy.

Now, I'd like to have your thoughts on another aspect of my problem 
which is basically a Stefan problem or a free or moving boundary 
problem, whatever you like to call it. I want to use a front-fixing 
method to alleviate the moving boundary difficulty. Basically, I do a 
coordinate transformation to get a non-dimensional space coordinate, 
which allows me to use regular boundary conditions. However, this 
results in an additional ODE that needs to be solved along with the PDEs.

Here is an example, not the problem I want to solve, but the idea is the 
same:

\[
\frac{\partial^2 u}{\partial \xi ^2} = s^2 \frac{\partial u}{\partial t}
- s \xi \frac{ds}{dt}\frac{\partial u}{\partial \xi}
\]

where $\xi$ is the non-dimensional space coordinate, $s = s(t)$ and
\[
-\frac{1}{s} \left. \frac{\partial u}{\partial \xi}\right|_{\xi = 1}
= \frac{ds}{dt}
\]

My first idea was to declare $s$ as a Variable() as in the diffusion 
example on page 64 of the manual, ``manually'' linearize $\frac{ds}{dt}$ 
with something like $\frac{s-s_{old}}{\Delta t}$ and solve the resulting 
algebraic equation for $s$ at every sweep.

Could you please give me your comments on that method? Perhaps you know 
of a better and/or more general approach.

Many thanks!

--

-- 
Etienne Rivard, MSc
Mitarbeiter
Lehrstuhl für Technische Thermodynamik und Transportprozesse (LTTT)

Postanschrift:
Universität Bayreuth
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95440 Bayreuth