View source for OpenFOAM guide/Discretization coefficients

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:

<math>A_{p} \boldsymbol\psi_{p} = \sum_r A_{r,p} \boldsymbol \psi_{r,p} + \mathbf{S}_p</math>

Where:

* ''p'' is the cell volume for which the equation applies;
* ''r'' are the ''related'' cell volumes (i.e. in close proximity to cell ''p'')<ref>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.</ref>;
* ''A'' are the discretization coefficients;
* <math>\boldsymbol \psi</math> 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:
* [[Matrix coefficients]]

== Notes ==
<references/>

[[Category:Numerical methods]]