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{\bf CME306 / CS205B Homework 2}
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\section*{Rotation Matrices}

\begin{enumerate}
\item The ODE that describes rigid body evolution is given by $R'=w^\star R$.

\begin{enumerate}
\item[(a)] Write down the forward Euler update for this equation.
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\item[(b)] Show that the updated ``rotation'' matrix computed from this update is not orthogonal (you may assume that $R^n$ {\em is} orthogonal).
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\item[(c)] How might one ``fix'' this matrix that is almost a rotation to make it rotation?
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\end{enumerate}

\item Another update equation is $R^{n+1} = e^{\Delta t (\omega^n)^\star} R^n$.  Here, $\omega^\star$ is the cross product matrix, where $\omega^\star v = \omega \times v$.  The exponential map $e^A$ can be defined for square matrices using the taylor series expansion
\[ e^{\Delta tA} = I + \Delta tA + \frac{\Delta t^2}{2} A^2 + \frac{\Delta t^3}{6}A^3 + ... \]

\begin{enumerate}
\item[(a)] Show that this update equation is a first order approximation of the ODE $R'=w^\star R$ that it is meant to solve.
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\item[(b)] Show that the result of this update is orthogonal.
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\item[(c)] Find a closed form expression for this update rule.  {\em (Hint: the cross-product matrix is closed under addition; i.e.\ $(\omega^\star)^3 = -|\omega|^2\omega^\star$... and don't forget your trigonometric identities!)}
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\item[(d)] Give an intuitive description of what this update rule is doing.
\end{enumerate}
\end{enumerate}

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\section*{Modified Equations}
\noindent Consider the advection equation
\begin{equation*}
u_t + a u_x = 0.
\end{equation*}
The numerical methods below satisfy \textit{modified equations} to higher order than the advection equation itself.
See Leveque \S 8.6 and the discussion notes for more on modified equations.  You should assume $\lambda = \frac{\Delta t}{\Delta x}$ is a constant.

\begin{enumerate}
\item[A.]
Find a modified equation for which explicit central differencing gives an $\bigo(\Delta t^2)$ approximation (we never want more than first-order derivatives in time, so if your solution contains $u_{tt}$, convert them to spatial derivatives).
What modification to the explicit central differencing scheme does this suggest to make it a stable numerical scheme for the advection equation?\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\

\item[B.]
Find a modified equation for which your proposed method from A gives an $\bigo(\Delta t^3)$ approximation.
\end{enumerate}
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