3.7 Laplacian discretisation

Let us first describe the discretisation of the Laplacian term for diffusion, introduced in Sec. 2.14 .

PICT\relax \special {t4ht=

Following the finite volume principles described in Sec. 3.1, the discretisation approximates the surface integral by a summation over faces by

pict\relax \special {t4ht=

The eqn 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 eqn and eqn of a matrix equation.

The mesh data eqn and eqn are calculated according to Sec. 3.3 so the remaining properties to be determined are:

  • the diffusivity at faces eqn ;
  • the surface normal gradient at faces eqn .

Fields eqn and eqn are associated with cells, so numerical schemes are required to evaluate properties at faces. We will first describe interpolation for eqn, for which the linear scheme is generally used. The surface normal gradient eqn 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 eqn for each face according to

f = w P + (1 w) N; \relax \special {t4ht=
where eqn is the interpolated face field. The subscripts eqn and eqn indicate values at owner and neighbour cells, respectively.

Linear interpolation

PICT\relax \special {t4ht=

The linear interpolation scheme sets eqn according to a linear variation between cells values eqn and eqn. 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

w = dfN- = --n---dN----: dPN n (dP + dN) \relax \special {t4ht=
Notes on CFD: General Principles - 3.7 Laplacian discretisation