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{\bf EGN 5456 \hfill Computational Mechanics \hfill 11/15/99}\\
{\it Closed book \hfill Van Dommelen \hfill 12:40-1:55pm}
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Consider the following scheme for the heat equation:
\begin{displaymath}
    \frac{u_{j}^{n+1}-u_{j}^{n}}{\Delta t} =
    \frac{u_{j+1}^{n}-u_{j}^{n}-u_{j}^{n+1}+u_{j-1}^{n+1}}
      {\left(\Delta x\right)^2}
\end{displaymath}

Analyze this scheme.  What is the molecule?  How would you compute a
time plane $n+1$ assuming that time plane $n$ is already known?  Under
what conditions will this scheme converge for all initial data?  What
is the accuracy?

If you believe in the CFL condition for stability, it implies the
scheme can only be stable for $\Delta t << \Delta x$.  Does that
agree with your results?

How would you rate this scheme compared to the standard FTCS for the
heat equation?

Show all reasoning and intermediate results leading to your answer.

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<BR><A HREF="figures/1.gif">Solution page 1.</A>
<BR><A HREF="figures/2.gif">Solution page 2.</A>
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