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{\bf EML 5725 \hfill Computational Fluid Dynamics \hfill 2/21/03}\\
{\it Closed book \hfill Van Dommelen \hfill 10:45-11:35 pm}
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\begin{enumerate}

\item
The following numerical scheme:
\begin{displaymath}
  \frac{u^{n+1}_{j}-2u^{n}_{j}+u^{n-1}_{j}}{(\Delta y)^2}
  + A \frac{u^{n+1}_{j+1}-2u^{n+1}_{j}+u^{n+1}_{j-1}}{(\Delta x)^2} = 0
\end{displaymath}
discretizes the Laplace equation when A = 1 and the wave equation when
A is negative.  Assume it is used to solve for $u^{n+1}_{j}$ when
u at lower values of $n$ has already been determined.  What does the CFL
condition say about this scheme?

\begin{rawhtml}
  <a href="figures/1.gif">Solution</a>
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\item
The Lax-Wendroff scheme is
\begin{displaymath}
  \left[
    \frac{\Delta_t }{\Delta t} + a \frac{\bar\delta_x}{\Delta x}
    - {\textstyle\frac12} a^2 \Delta t \frac{\delta_x^2}{(\Delta x)^2}
  \right] u^{n}_{j} = 0
\end{displaymath}
Write this scheme out in terms of mesh point values of $u$.
Find the PDE which which this scheme is consistent.
Find the normal accuracy of this scheme and any special conditions
under which the scheme would be more accurate.

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  <a href="figures/2.gif">Solution</a>
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\item
Consider the function $u=\cos(x)$ for $0\le x\le\pi$.
Using $\Delta x = \pi/2$, approximate the derivative of
$u$ at $x=\pi/2$ using both the second order accurate central
difference scheme, as well as the fourth order accurate compact
difference scheme.   Which scheme is more accurate?

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  <a href="figures/3.gif">Solution</a>
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\end{enumerate}

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