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{\bf CME306 / CS205B Homework 6}
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\section*{Essentially Non-Oscillatory Schemes}
Given the following data for $\phi^n$, write down the interpolating polynomial that third order HJ ENO would construct 
in order to compute $\phi_i^{n+1}$ in approximating the equation $\phi_t + \phi_x = 0$.
\begin{displaymath}
\phi_{i-3}^n = 5, \phi_{i-2}^n = 5, \phi_{i-1}^n = 4, \phi_{i}^n = 5, \phi_{i+1}^n = 1,  \phi_{i+2}^n = -2, \phi_{i+3}^n = 0
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\pagebreak
\section*{Weighted ENO}
If we consider an upwind discretization of $\phi_x$, we have three possible third-order interpolating polynomials, given by 
\begin{align*}
\phi^1_x &= \frac{v_1}{3} - \frac{7v_2}{6} + \frac{11v_3}{6} \\
\phi^2_x &= -\frac{v_2}{6} + \frac{5v_3}{6} + \frac{v_4}{3} \\
\phi^3_x &= \frac{v_3}{3} + \frac{5v_4}{6} - \frac{v_5}{6}
\end{align*}
Where $v_j=D^*\phi_{i+j-3}$, and $D^*\phi$ is the first-order upwind discretization of $\phi_x$.

However, the philosophy of picking exactly one of the three candidate stencils is overkill in smooth regions of $\phi$ where $\phi$ is well-behaved.  Instead, we can take a convex sum of the three stencils,
\begin{equation}
\phi_x = \omega_1\phi^1_x + \omega_2\phi^2_x + \omega_3\phi^3_x
\end{equation}
Where $0 \leq \omega_i \leq 1$, $\omega_1+\omega_2+\omega_3 = 1$.  It has been shown that we can pick $\omega_1=.1,\omega_2=.6,\omega_3=.3$ and achieve a $5^{th}$ order accurate approximation of $\phi_x$.

\begin{enumerate}
\item Show that if we perturb $\omega$ by $\bigo(\Delta x^2)$ we still get a $5^{th}$ order approximation to $\phi_x$. \\ \\ \\ \\ \\ \\ \\ \\ \\ \\
\item Why is this a bad idea in non-smooth areas of the flow?  In order to demonstrate this, consider $\phi_t+\phi_x=0$ for a heaviside step function, with initial data given by:
\begin{displaymath}
\phi_{i-3}^n = 0, \phi_{i-2}^n = 0, \phi_{i-1}^n = 0, \phi_{i}^n = 1, \phi_{i+1}^n = 1,  \phi_{i+2}^n = 1, \phi_{i+3}^n = 1
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