The surface normal gradient is a part of the Laplacian discretisation Eq. (3.2 ), illustrated in the ﬁgure below. The discretisation of is built upon a ﬁnite diﬀerence of cell values on each side of the face according to (3.5)
where . When this orthogonal scheme is applied to Eq. (3.2 ) to discretise a Laplacian, it forms coeﬃcients of a matrix equation since it references cell values of the ﬁeld . For cell , the coeﬃcient for each neighbour cell ( ) is and the diagonal coeﬃcient is the negative of the sum of neighbour coeﬃcients: .

Discretisation of by Eq. (3.5) is most accurate when the face is orthogonal to , i.e. the angle between and is zero. However, if the face is non-orthogonal, the error associated with Eq. (3.5 ) increases with .

### Non-orthogonal correction

A more accurate discretisation of at a non-orthogonal face is formed of the vector sum of the orthogonal scheme and an explicit correction . The latter is calculated from the full gradient in adjacent cells (described in Sec. 3.15 ), interpolated to the face . The correction is explicit, i.e. calculated using known values of , so may need updating within an iterative sequence to maintain accuracy, as discussed in Sec. 5.20 . To ensure that the iterative sequence converges, the implicit contribution is elevated by replacing in the orthogonal scheme with (3.6)
The corrected scheme combines the implicit and explicit parts by (3.7)
The corrected scheme is generally stable for . For , stability can be maintained at the expense of accuracy by limiting the magnitude of the correction below some fraction of the magnitude of the orthogonal part.
Notes on CFD: General Principles - 3.8 Surface normal gradient 