3.8 Surface normal gradient
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
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 .
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