Where, \(\overline{\nabla\Phi} V\) is the average gradient over the volume V.

Φf for all the faces should be known to compute ∇ΦC.

∇Φf=gC ∇ΦC+gf ∇Φf.

Explanation: The face-based computation of Φf uses a compact stencil involving face neighbours. This is less accurate than the vertex-based computation involving a large stencil of vertex neighbours.

Explanation: Compact stencil uses implicit methods. So, it is easy to construct compact Jacobian matrices. But the large stencil brings more information into the reconstruction and therefore is more accurate.

gC=\(\frac{distance_{Ff}}{distance_{FC}}\).

Explanation: The face-centred stencil is second-order accurate when the centroid of the face is the same as the point of intersection of the face and the line connecting the two cells. This happens when the centroid and the line connecting the two cells meet.

Φf=\(\frac{\phi_{n1}+\phi_{n2}}{2}\).

Explanation: The impact decreases as the centroid of the cell moves away from the vertex. So, the inverse is used as the weight here. This is why the method needs an enlarged stencil. But it leads to more accurate approximation.