3.7 Laplacian discretisation
Let us ﬁrst describe the discretisation of the Laplacian term for diﬀusion, introduced in Sec. 2.14 .
Following the ﬁnite volume principles described in Sec. 3.1, the discretisation approximates the surface integral by a summation over faces by
The in Eq. (3.2 ) can be viewed as the summation over faces of a single cell. Applying the summation to all cells provides the contribution to coeﬃcients and of a matrix equation.
The mesh data and are calculated according to Sec. 3.3 so the remaining properties to be determined are:
- the diﬀusivity at faces ;
- the surface normal gradient at faces .
Fields and are associated with cells, so numerical schemes are required to evaluate properties at faces. We will ﬁrst describe interpolation for , for which the linear scheme is generally used. The surface normal gradient is discussed in Sec. 3.8 .
Interpolation from cells to faces
Mapping data between diﬀerent locations is a common practice in numerics. Since the ﬁnite volume method is concerned with ﬂuxes at faces, the principal mapping procedure is interpolation from cells to faces. Since other interpolations are much less common, it can be assumed that the term “interpolation” means from cell to face unless stated otherwise.
For irregular polyhedral meshes, interpolation is generalised by deﬁning a weights ﬁeld for each face according to
The linear interpolation scheme sets according to a linear variation between cells values and . The weights can then be calculated based on distances from the face centre to adjacent cell centres, in the direction normal to the face, by