\documentclass{article}
\usepackage{fullpage}
\usepackage{amsmath}
\usepackage{graphicx}
\usepackage{pstricks}
\usepackage{pst-plot}
\usepackage{subfigure}

\newcommand{\bigo}{\ensuremath{\mathcal{O}}}
\newcommand{\pderiv}[2]{\ensuremath{\frac{\partial #1}{\partial #2}}}

\begin{document}
\begin{center}
{\bf CME306 / CS205B Homework 2}
\end{center}

\section*{Arbitrary Lagrangian-Eulerian (ALE) Methods}
Recall from homework that we derived the weak form of conservation of mass (in Eulerian form) to be:
\begin{equation}
\pderiv{}{t}\int_\Omega \rho dV + \int_{\partial\Omega}(\rho\vec{u})\cdot \vec{dA} = 0
\end{equation}
Where $\Omega$, a control volume, remains fixed in time.  In Lagrangian methods, we instead move $\Omega$ and ignore the flux across the boundary.  ALE methods make no such assumption, and instead we take the change in time of the boundary to be $\pderiv{\Omega}{t} = \vec{v} \neq \vec{u}$.
\begin{enumerate}
\item Please re-derive the weak form of conservation of mass, this time in ALE form (that is, the control volume $\Omega$ is moving at some speed $\vec{v}$, which is not the fluid velocity $\vec{u}$).  Remember that conservation of mass describes the change in mass of a control volume, so $\pderiv{}{t}$ should {\em not} be under the volume integral.  \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\

\item Write down the strong form of conservation of mass, in ALE form.
\end{enumerate}

\pagebreak
\section*{Runge-Kutta methods}
Recall the model ordinary differential equation, $y' = \lambda y$, can be discretized and solved in a variety of ways.  A popular family of methods are referred to as RK, or Runge-Kutta methods (you may recall that the first order RK method is equivalent to forward-differencing, $y_{i+1} = y_i + \Delta x\lambda y_i$).  These methods can be expressed generally as $y_{i+1} = Gy_i$, and are stable when $|G| \leq 1$ -- this gives a condition on $\Delta x\lambda$ for stability.
\begin{enumerate}
\item {\em TVD}---Define the `total variation' of $v$ as
\begin{equation}
TV(v) = \sum_{j=1}^n\left|v_{j+1} - v_j\right|
\end{equation}
And prove that $2^{nd}$ order Runge-Kutta is total variation diminishing (TVD) in the sense that $TV(v^{n+1}) \leq TV(v^n)$.  You should assume that forward Euler is TVD.  Recall that $2^{nd}$ order Runge-Kutta is given to be:
\begin{equation}\begin{cases}
y^* &= \left( 1 + \Delta x\lambda \right)y_i \\
y^{**} &= \left( 1 + \Delta x\lambda \right)y^* \\
y_{i+1} &= \frac{y_i + y^{**}}{2}
\end{cases}\end{equation}\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ 

\item Note that $\lambda$ in general can be complex, and find the stability condition for $2^{nd}$ order Runge-Kutta.
\end{enumerate}

\pagebreak
\section*{Lax-Richtmyer Theorem}
Prove that stability and consistency are sufficient for convergence for a linear scheme.

\end{document}
