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\title{CS 237C(CME 306) \\ Midterm I}
\date{May 2, 2005}
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\begin{center}
\begin{tabular}{|l|p{3in}|}
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Name & \\
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SUID & \\
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Signature & \\
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\begin{tabular}{|l|l|p{.4in}|}
\hline Question & Points & Score \\
\hline 1 & 15 & \\
\hline 2 & 5 & \\
\hline 3 & 5 & \\
\hline 4 & 5 & \\
\hline 5 & 5 & \\
\hline 6 & 5 & \\
\hline 7 & 5 & \\
\hline 8 & 5 & \\
\hline 9 & 5 & \\
\hline 10 & 25 & \\
\hline 11 & 20 & \\
\hline Total & 100 & \\
\hline 
\end{tabular}

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\begin{enumerate}

\item Indicate whether each statement is true or false.
    \begin{description}
    \item[T / F] Unlike shocks, contact discontinuities travel at the local characteristic speed.
    \item[T / F] The weak solution of a conservation law is unique.
    \item[T / F] The vanishing viscosity solution of a conservation law converges to the entropy satisfying weak solution of the conservation law.
    \item[T / F] Consider a weak solution for Burgers equation.  In the interior of a rarefaction wave, the solution also satisfies the strong form.
    \item[T / F] A high resolution method refers to a method where, almost everywhere in the domain, the dominant error term is high order, even if the method has a low order of accuracy.
    \item[T / F] A linear, second order accurate numerical method is stable if and only if it is convergent.
    \item[T / F] The Lax-Wendroff theorem tells us under which conditions a numerical approximation can be expected to converge.
    \end{description}

\item Let $\cal{N}$ denote a consistent, linear, one-step numerical method.  Which of the following statements are true?
    \begin{enumerate}[labelindent=\parindent, leftmargin=*, label=\Roman*., align=left]
    \item If $\left \Vert \cal{N} \right \Vert > 1$  then the method is unstable.
    \item If the method yields a numerical solution whose norm grows exponentially in time then the method is unstable.
    \item If the method does not satisfy the CFL condition then the method is unstable.
    \end{enumerate}

    \begin{enumerate}
    \item I only
    \item II only
    \item III only
    \item II and III only
    \item I, II, and III
    \end{enumerate}

\pagebreak

\item The ENO-RF method is preferable to the ENO-Roe method for computing the numerical flux function because ENO-Roe
    \begin{enumerate}
    \item results in too much dissipation in the numerical solution.
    \item may create spurious oscillations in the vicinity of shocks and other nonlinear phenomena.
    \item may compute a nonphysical weak solution.
    \item has a lower order of accuracy.
    \item can be shown to diverge in some cases where ENO-RF converges.
    \end{enumerate}

\item Which statement is true?
    \begin{enumerate}
    \item Second order TVD Runge-Kutta is an improvement upon second order Runge-Kutta because the latter is not always TVD.
    \item The exact solution of the scalar, constant coefficient advection equation has total variation which decreases in time. Therefore, TVD schemes which mimic this behavior are desirable.
    \item If we have a spatial discretization such that a forward Euler step is TVB, then third order TVD Runge-Kutta in conjunction with that spatial discretization is TVB.
    \item If we have a spatial discretization such that a forward Euler step is TVB, then third order TVD Runge-Kutta in conjunction with that spatial discretization is TVD.
    \item None of the above.
    \end{enumerate}

\item Consider the first order upwind method for the scalar, constant coefficient advection equation $u_t + a u_x = 0$.  Which statement is true?
    \begin{enumerate}
    \item Modified equation analysis for the method indicates that the numerical solution will exhibit dispersive behavior.
    \item In exact arithmetic, if we take $\frac{\triangle x}{\triangle t} = \left| a \right|$, the method can yield the exact solution.
    \item The method is unconditionally stable.
    \item The method is preferable to explicit central differencing, although both methods have the same order of accuracy.
    \item None of the above.
    \end{enumerate}

\pagebreak

\item Consider the scalar conservation law $u_t + \left( \frac{u^2}{2} \right)_x = 0$.  Which of the following statements are necessarily true?
    \begin{enumerate}[labelindent=\parindent, leftmargin=*, label=\Roman*., align=left]
    \item The characteristic curves are straight lines.
    \item Shocks travel in straight lines.
    \item Shocks cannot form if the initial data is smooth.
    \end{enumerate}

    \begin{enumerate}
    \item I only
    \item II only
    \item III only
    \item I and II only
    \item I, II, and III
    \end{enumerate}

\item Which of the following schemes for Burgers equation can be written in discrete conservation form?
    \begin{enumerate}[labelindent=\parindent, leftmargin=*, label=\Roman*., align=left]
    \item $\frac{u^{n+1}_j-u^n_j}{\triangle t} + u^n_j \left( \frac{u^n_{j+1}-u^n_j}{\triangle x} \right) = 0$
    \item $\frac{u^{n+1}_j-u^n_j}{\triangle t} + u^n_j \left( \frac{u^n_{j+1}-u^n_{j-1}}{2 \triangle x} \right)= 0$
    \item $\frac{u^{n+1}_j-u^n_j}{\triangle t} + \frac{1}{\triangle x} \left( \frac{1}{2}\left(u^n_{j}\right)^2-\frac{1}{2}\left(u^n_{j-1}\right)^2 \right) = 0$
    \end{enumerate}
    
    \begin{enumerate}
    \item None 
    \item III only
    \item II and III only
    \item I and II only
    \item I and III only
    \end{enumerate}

\item ENO schemes construct an interpolating polynomial in order to
    \begin{enumerate}
    \item use higher order interpolation near steep gradients and discontinuities.
    \item create a numerical stencil that avoids crossing shocks.
    \item allow for higher order in time approximations to $u_t$.
    \item remain conservative.
    \item none of the above
    \end{enumerate}

\pagebreak 

\item Consider a consistent, stable, linear, one-step numerical method.  Which of the following statements are true?
    \begin{enumerate}[labelindent=\parindent, leftmargin=*, label=\Roman*., align=left]
    \item Local truncation errors accumulate at each time step.
    \item Local truncation errors accumulate and may be amplified at each time step.
    \item In order to get convergence to the exact solution, we must choose the initial data for the scheme to be equal pointwise to the initial data for the exact solution.
    \end{enumerate}

    \begin{enumerate}
    \item I only
    \item III only
    \item I and II only
    \item I and III only
    \item I, II and III
    \end{enumerate}
    
\pagebreak
\item  Consider the scalar, constant coefficient advection equation
\begin{displaymath}
    u_t + a u_x = 0
\end{displaymath}

and the following numerical method for computing an approximate solution to it.

\begin{displaymath}
    u^{n+1}_j = \frac{1}{2}\left( u^n_{j-1} + u^n_{j+1} \right) - \frac{\triangle t}{2\triangle x}a \left( u^n_{j+1} - u^n_{j-1} \right)
\end{displaymath}

Analyze the stability and consistency of the method.  What is the order of accuracy of the method?
What can you say about convergence of the method?
You may assume that $\frac{\triangle t}{\triangle x} = constant$.

\pagebreak

\pagebreak
\item Consider the scalar, constant coefficient advection equation
\begin{displaymath}
    u_t + a u_x = 0
\end{displaymath}

Show that for this equation, the ENO-Roe and ENO-LLF schemes will produce identical numerical approximations.




\end{enumerate}


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