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\title{Homework 3}

\markboth{CS 237c, Spring 2006} {Shell \MakeLowercase{\textit{et
al.}}: Bare Demo of IEEEtran.cls for Journals} \maketitle

\section{Modified Equations}
Consider the advection equation
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u_t + a u_x = 0.
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The numerical methods below satisfy \textit{modified equations} to higher order than the avdection equation itself.
See Leveque \S 8.6 for more on modified equations.
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Assume $\lambda = \frac{\triangle t}{\triangle x} = constant$.

\subsection{Explicit central differencing}
Find a modified equation for which explicit central differencing gives an $O(\triangle t^2)$ approximation.
What modification to the explicit central differencing scheme does this suggest to make it a stable numerical scheme for the advection equation?  What is the resulting scheme called?

\subsection{Lax-Wendroff}
Find a modified equation for which the Lax-Wendroff method gives an $O(\triangle t^3)$ approximation.

% that's all folks
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