OpenFOAM guide/Discretization coefficients

From OpenFOAMWiki

Discretization coefficients are coefficients that result from the discretization of the problem. They are closely related to the matrix coefficients. OpenFOAM uses matrix coefficients in its source code.

The discretization coefficients have the form:

A_{p} \boldsymbol\psi_{p} = \sum_r A_{r,p} \boldsymbol \psi_{r,p} + \mathbf{S}_p


  • p is the cell volume for which the equation applies;
  • r are the related cell volumes (i.e. in close proximity to cell p)[1];
  • A are the discretization coefficients;
  • \boldsymbol \psi is the variable being solved for; and
  • S is the source term.

The neighbour cells, indexed by r, depend on the size of the computational molecule, which depends on the discretization scheme. For simple discretization schemes, the neighbour cells only includes adjacent cells; whereas for higher order schemes, more distant cells are included in the summation. In orthogonal grids, the notation for the neighbours is usually expressed in terms of compass directions; however this cannot be used in unstructured grids.

See also:


  1. What I'm calling related cells is conventionally called neighbours. But OpenFOAM has a different meaning for neighbours, so the term related cells is used.