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

  $$P^k(x,y,z)=\sum^k_{i=0} \sum^{k-i}_{j=0} \sum^{k-(i+j)}_{l=0} C_{i,j,l}x^i y^j z^l$$

  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. P¹ (x,y,z) = C₀+C₁ x+C₂ y+C₃ 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 P¹ above

  $$\left [ \begin{array} {cccc} 1 & \overline{x}_1 & \overline{... ...Q}_2 \\ \overline{Q}_3 \\ \overline{Q}_4 \\ \end{array}\right )$$