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\begin{center}
	{\bf $k$-exact Reconstruction}
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\begin{itemize}
\item The problem of determining the point-wise variation of the flow
variables from the cell averages is known as the reconstruction problem.

\item Simply stated, the reconstruction problem involves the process of
determining the point-wise distribution given the cell averages with the
restriction that integrating the point-wise function recovers the cell
averages.

\item The variation of the flow quantities is approximated by a
polynomial over each control volume.

\item The polynomial satisfies a criteria called $k$-exactness which
states that the polynomial reconstruction be exact i.e. recovers the
cell average correctly whenever a polynomial of degree $k$ is used.

\item A general polynomial of degree $k$ in 3D is written as 
$$
P^k(x,y,z)=\sum^k_{i=0} \sum^{k-i}_{j=0} \sum^{k-(i+j)}_{l=0}
C_{i,j,l}x^i y^j z^l
$$
In 3D, the polynomial contains $\frac{(k+1)(k+2)(k+3)}{6}$ coefficients
and in 1D and 2D, it contains $k+1$ and $\frac{(k+1)(k+2)}{2}$ coefficients
respectively. e.g. $P^1 (x,y,z) = C_0+C_1 x+C_2 y+C_3 z$

\item As a final task to complete the reconstruction, a support stencil
of number equal to the number of coefficients is selected, and the resulting
linear system solved. e.g for $P^1$ above
$$
\left [
\begin{array}{cccc}
1 & \overline{x}_1 & \overline{y}_1 & \overline{z}_1 \\
1 & \overline{x}_2 & \overline{y}_2 & \overline{z}_2 \\
1 & \overline{x}_3 & \overline{y}_3 & \overline{z}_3 \\
1 & \overline{x}_4 & \overline{y}_4 & \overline{z}_4 \\
\end{array}
\right ]
\left (
\begin{array}{c}
C_0 \\
C_1 \\
C_2 \\
C_3 \\
\end{array}
\right )
=
\left (
\begin{array}{c}
\overline{Q}_1 \\
\overline{Q}_2 \\
\overline{Q}_3 \\
\overline{Q}_4 \\
\end{array}
\right )
$$


\end{itemize}

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