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\centerline{Workshop \#7, diffusion, outline for TF, not students}
\centerline{APM 115: mathematical modeling}

\bigskip
2D Diffusion with no-flux boundary conditions.
\bigskip

To the students: 
\begin{itemize}
\item What is the physical meaning of a no-flux boundary condition?
\item Change the program \texttt{diffusion\_demo\_apm115.m} to use
  zero flux boundary conditions instead of specified temperature on
  the boundaries.
\item Calculate and plot a time series of the spatially averaged
  temperature.
\item Run all three demos and explain the results.  In particular,
  what is the steady state of the cases with non zero sources?
\end{itemize}

To the TF: 
\begin{itemize}
\item No flux boundary conditions correspond to insulating
  boundaries.  
\item Start by showing the students the solution to the no flux
  boundary conditions as compared with the fixed temperature boundary
  conditions using the Matlab scripts workshop07.m and the one used in
  class.  Ask what the difference is\ldots
\item Help the students understand that they need to solve the
  equation at boundary points too now, not just interior points as
  done in the demo used in class.
\item They need to do this by calculating the $Flux_{in}$ and
  $Flux_{out}$ terms at the boundaries too.
\item So for example, the in-x-flux is zero on the left boundary, and
  the out-y-flux is zero on the top boundary.
\item Explain to the students that if we calculate the time derivative
  of the spatial average of the temperature,
  \[
  \overline{T}^n\equiv\sum_{i,j}T_{i,j}^n,
  \]
  then the sum over the $Flux_{in}$ terms (see code) creates a
  telescopic series which sums to zero with zero flux boundary
  conditions, and that therefor the spatially averaged temperature is
  constant in time unless there is a source.  If there is a source,
  the averaged temperature increases linearly in time at a rate equal
  to the spatial average of the source.
\end{itemize}
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