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{\bf EGN 5456 \hfill Computational Mechanics \hfill 10/30/98}\\
{\it Closed book \hfill Van Dommelen \hfill 8:35-9:25 am}
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Show all reasoning and intermediate results leading to your answer.

The following PDE describes convection of entropy in a pipe:
\begin{displaymath}
  {\partial s\over\partial t} + u {\partial s\over\partial x} = 0
\end{displaymath}
where $s=s(x,t)$ is entropy, $t$ time, $u>0$ a given constant flow
velocity in the pipe, and $x$ the distance along the pipe.
The boundary and initial conditions are:
\begin{displaymath}
  s(0,t) = 1 \qquad s(x,0)=0
\end{displaymath}
The length of the pipe is 2.

We want to solve this problem numerically using the following scheme:
\begin{displaymath}
  {s^{n+1}_j - s^n_j\over \Delta t}
  + {u\over 2} {s^{n+1}_j - s^{n+1}_{j-1}\over\Delta x}
  + {u\over 2} {s^n_{j+1} - s^n_j\over\Delta x} = 0
\end{displaymath}
where $s_j^n$ is the value of the entropy at mesh point number $j$
and time step number $n$, $\Delta t$ the time increment, and $\Delta x$
the spacing between mesh points.

Discuss this scheme in as much detail as possible, discussing the
following questions fully (in any order):
\begin{enumerate}
\item Does the scheme make sense? (25\%)
\item Will the scheme work? (25\%)
\item How well will the scheme work? (25\%)
\item Exactly how would you need to perform the computation? (25\%)
\end{enumerate}

To save time, in case you feel the need to do any Taylor series
expansions, you may represent terms involving third order derivatives
and higher by order symbols.

\strut
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<A HREF="figures/1.gif">Solution page 1</A>
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<A HREF="figures/2.gif">Solution page 2</A>
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<A HREF="figures/3.gif">Solution page 3</A>
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