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{\bf CME306 / CS205B Homework 9}
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\begin{enumerate}
\item Write out the {\em symmetric} matrix equation for the standard second order central difference approximation to the equation
\begin{equation}
  \nabla\cdot\left(\frac{1}{\rho}\nabla p\right) = \nabla\cdot\vec{u}
\end{equation}
with the following boundary conditions:
\begin{equation}
  \begin{cases}
    p(0,y) = 1 &\\
    p(1,y) = 1 &\\
    p_y(x,y) = 0 & \mbox{for } y\in\{0,1\}
  \end{cases}
\end{equation}
You should assume a MAC grid (ie. that velocities live on cell faces, and that pressure and density live in the cell centers), and you may {\em not} assume a constant density.  Write the equations for the following three cells:
\begin{enumerate}
\item an internal cell (something sufficiently far from the boundary, ie. $p_{ij}$)
\item a cell that lies along the x-axis boundary (ie. $p_{i1}$), and
\item a cell that lies along the y-axis boundary (ie. $p_{1j}$).
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\end{enumerate}
\item Physically, when we have an incompressible flow with all Neumann boundary conditions, what does the compatibility condition require?  Is something similar required for compressible flow?
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