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\begin{center}
{\bf EML 5725 \hfill Computational Fluid Dynamics \hfill 4/2/03}\\
{\it Closed book \hfill Van Dommelen \hfill 10:45-11:35 pm}
\end{center}

\begin{enumerate}

\item The MAC scheme uses staggered locations of the velocity
  components $u$ and $v$ as shown below.  Thus the continuity equation
  ${\rm div} \vec v=0$ can be approximated to second order using the
  following simple formula:
  \begin{displaymath}
    \frac{u_{j+1,l}-u_{j,l}}{\Delta x} +
    \frac{v_{j,l+1}-v_{j,l}}{\Delta y} = 0
  \end{displaymath}
  Here $j$ is the mesh line index in the horizontal direction and
  $l$ the one in the vertical direction.
  Determine whether this formula is conservative.

\begin{rawhtml}
  <a href="figures/1.gif">Solution</a>
\end{rawhtml}

  \begin{displaymath}
    \epsffile{figures/m.ps}
  \end{displaymath}

\item Consider the {\em nonuniform} mesh shown below,
  where $j$ is the mesh line index in the horizontal direction and
  $l$ the one in the vertical direction.  The mesh spacings are,
  by definition,
  $\Delta x_j \equiv x_{j+1}-x_j$ and
  $\Delta y_l \equiv y_{l+1}-y_l$.
  Derive a {\em second order} accurate formula for the derivative
  $\partial T/\partial y$ at an arbitrary wall point $j,0$ using
  the values of the temperature $T_{j,0}$, $T_{j,1}$, and $T_{j,2}$
  and the mesh spacings.  Show the second order accuracy.

\begin{rawhtml}
  <a href="figures/2.gif">Solution</a>
\end{rawhtml}

  \begin{displaymath}
    \epsffile{figures/b.ps}
  \end{displaymath}

\item Determine the stability of the following scheme:
\begin{displaymath}
    \frac{\delta_t}{\Delta t} u_{j}^{n+\frac12} +
    c \frac{\nabla_x}{\Delta x} \mu_t u_{j}^{n+\frac12} = 0
\end{displaymath}

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

\end{enumerate}

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