k-exact Reconstruction

• The problem of determining the point-wise variation of the flow variables from the cell averages is known as the reconstruction problem.

• 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.

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

• 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.

• A general polynomial of degree k in 3D is written as

P^{k}(x, y, z) = k ; ∑ ; i=0 k−i ; ∑ ; j=0 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

• 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

[ \begin{matrix}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{matrix} ] ( \begin{matrix}C_{0} \\ C_{1} \\ C_{2} \\ C_{3} \end{matrix} ) = ( \begin{matrix}\overline{Q} & 1 \\ \overline{Q} & 2 \\ \overline{Q} & 3 \\ \overline{Q} & 4 \end{matrix} )