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\hyphenation{op-tical net-works semi-conduc-tor}
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\title{Homework 1}

\markboth{CS 237c/CME 306} {Shell \MakeLowercase{\textit{et
al.}}: Bare Demo of IEEEtran.cls for Journals} \maketitle

\section{Convergence Analysis}
Consider the wave equation:
\begin{displaymath}
u_t+au_x=0.
\end{displaymath}
Establish whether or not the following methods for solving the
equation converge. If so, what are the conditions for convergence.
\textbf{Hint}: Use the Lax-Richtmyer equivalence theorem.  Chapters
1 and 2 of the text by Strikwerda will be helpful.

\subsection{Explicit Central Differencing}
\begin{displaymath}
\frac{v^{n+1}_{j}-v^n_{j}}{{\Delta}t}+a\frac{v^{n}_{j+1}-v^n_{j-1}}{2{\Delta}x}=0.
\end{displaymath}

\subsection{Implicit Central Differencing}
\begin{displaymath}
\frac{v^{n+1}_{j}-v^n_{j}}{{\Delta}t}+a\frac{v^{n+1}_{j+1}-v^{n+1}_{j-1}}{2{\Delta}x}=0.
\end{displaymath}

\subsection{Upwinding}
\begin{displaymath}
\frac{v^{n+1}_{j}-v^n_{j}}{{\Delta}t}+aD^{*}v^{n}_j=0
\end{displaymath}

If $a>0$, $D^{*}=D^{-}$. If $a<0$, $D^{*}=D^{+}$.

\subsection{Downwinding}
\begin{displaymath}
\frac{v^{n+1}_{j}-v^n_{j}}{{\Delta}t}+aD^{*}v^{n}_j=0
\end{displaymath}

If $a>0$, $D^{*}=D^{+}$. If $a<0$, $D^{*}=D^{-}$.

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