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\begin{document}
\begin{center}
{\bf CME306 Qualifying Exam}
\end{center}
\section*{Part I - Multiple Choice (1 point each)}
\begin{enumerate}
\item If we have a spring with drag coefficient $k_d$ and spring constant $k_s$, which of the following are sufficient to have a well-posed system?
\begin{enumerate}
\item[(a)] $k_d > 0$
\item[(b)] $k_s > 0$, $k_d > 0$
\item[(c)] $k_sk_d < 0$
\item[(d)] $\left(\frac{k_d}{2m}\right)^2 - \frac{k_s}{mx_0} \geq 0$
\end{enumerate}

\item Suppose that we wish to discretize the equation
\begin{displaymath}
u_t - u_x = 0.
\end{displaymath}
Choose the best discretization among the following choices.
    \begin{enumerate}
    \item \begin{equation*}
        \frac{v^{n+1}_i-v^n_i}{\Delta t}-\frac{v^n_{i+1} - 2v^n_{i} + v^n_{i-1}}{\Delta x^2}=0
        \end{equation*}
    \item \begin{equation*}
        \frac{v^{n+1}_i-v^n_i}{\Delta t}-\frac{v^n_{i+1} - v^n_{i}}{\Delta x}=0
        \end{equation*}
    \item \begin{equation*}
        \frac{v^{n+1}_i-v^n_i}{\Delta t}-\frac{v^n_{i+1} - v^n_{i-1}}{2\Delta x}=0
        \end{equation*}
    \item \begin{equation*} \begin{cases}
        \frac{\hat{v}^{n+1}_i-v^n_i}{\Delta t}-\frac{v^n_{i+1} - v^n_{i-1}}{2\Delta x}=0 \\
        \frac{\hat{v}^{n+2}_i-\hat{v}^{n+1}_i}{\Delta t}-\frac{\hat{v}^{n+1}_{i+1} - \hat{v}^{n+1}_{i-1}}{2\Delta x}=0\\
        v^{n+1}_i = \frac{\hat{v}^{n+2}_i + v^n_i}{2}
        \end{cases}\end{equation*}
    \item \begin{equation*}
        \frac{v^{n+1}_i-v^n_i}{\Delta t}-\frac{v^n_{i} - v^n_{i-1}}{\Delta x}=0
        \end{equation*}
    \end{enumerate}        
\end{enumerate}
\pagebreak
\section*{Part II - Short answer}
\begin{enumerate}
\item (2 points) Please discuss briefly the advantages and disadvantages of using forward- vs.\ backward-Euler time-stepping.
\\ \\ \\ \\ \\ \\ \\ \\
\item (2 points) Why does Lax-Richtmyer require stability in addition to consistency (i.e.\ why isn't consistency sufficient)?
\\ \\ \\ \\ \\ \\ \\ \\
\item (2 points) Consider a simple equilateral triangle, with side lengths $\ell_{1_0}=\ell_{2_0}=\ell_{3_0}=1$.  In world space, the
  sides measure $\ell_1,\ell_2$ and $\ell_3$ respectively.  Write down the Green strain for this deformation (it is sufficient to write down $D_m$ and $D_m^TGD_m$).  \\ \\ \\ \\ \\


\end{enumerate}
\pagebreak
\section*{Part III - Long Answer}
\begin{enumerate}
\item (4 points)
\begin{equation}
u_t + au_x = 0 \label{eqn:advection}
\end{equation}
Show that the following discretization of the advection equation~\eqref{eqn:advection} with $a>0$ is either stable or unstable, then {\bf state} the order of accuracy (ie.\ there is no need to justify the order of accuracy).
\begin{equation}
\begin{cases}
u^*_i = u^n_i - a\Delta t\frac{3u^n_i - 4u^n_{i-1} + u^n_{i-2}}{2\Delta x} \\
u^{**}_i = u^*_i - a\Delta t\frac{3u^*_i - 4u^*_{i-1} + u^*_{i-2}}{2\Delta x} \\
u^{n+1}_i = \frac{u^{**}_i+u^n_i}{2}
\end{cases}
\end{equation}

\end{enumerate}
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