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{\bf EGN 5456 \hfill Computational Mechanics \hfill 11/2/01}\\
{\it Closed book \hfill Van Dommelen \hfill 1:05-1:55pm}
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Consider the following Van Dommelen scheme:
\begin{displaymath}
    \frac{u_{j}^{n+1}-u_{j}^{n}}{\Delta t} = \alpha
    \frac{u_{j+1}^{n}-2 u_{j}^{n+1}+u_{j-1}^{n}}
      {\left(\Delta x\right)^2}
\end{displaymath}

Tell me all you can about 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?  What
is the accuracy?  Will there be initial data for which the scheme will
not converge?

Use operators only, no Taylor series, and no $E$ operators either.
(Hint: write the left hand side as the ordinary form of that derivative
plus correction terms.  Each can be represented by non-$E$ operators.)

Compare the Van Dommelen scheme to the standard FTCS scheme for the
heat equation.  What are its disadvantages?  Exactly how bad are they?
What are its strong points?  Don't forget who will grade this test.

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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