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\title{Homework 2}

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

\section{Lax-Richtmyer Theorem}
\noindent Recall the standard wave equation,
\begin{displaymath}
u_t+au_x=0,
\end{displaymath}
where $a=constant$. Prove that stability and consistency are sufficient for convergence for approximations to the wave 
equation. See Chapter 8 of \emph{Finite Volume Methods for Hyperbolic Problems}, by Randall J. LeVeque (there is a link 
to an electronic copy of the book on the class website). 
\section{TVD}
\noindent Define the 'total variation' of $v$ as
\begin{displaymath}
TV(v)=\sum_{j=1}^{N}\left|{v_{j+1}-v_j}\right|.
\end{displaymath}
Prove that second order Runge-Kutta is total variation diminishing 
in the sense that $TV(v^{n+1})<=TV(v^n)$.  You may assume that the 
forward Euler step is TVD.

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