View source for OpenFOAM guide/Matrix coefficients

The matrix coefficients are coefficients that result from the [[discretization]] process.  They are closely related to the [[discretization coefficients]].  [[OpenFOAM]] uses matrix coefficients in its source code.

== Definition ==
The matrix coefficients have the form:

<math>A_{ii} \boldsymbol \psi_i + \sum_j A_{ij} \boldsymbol \psi_j = \mathbf{S}_i</math>

or:

<math>\mathbf{A}\boldsymbol \psi = \mathbf S</math>

Where:

*''A'' are the matrix coefficients;
* <math>\boldsymbol \psi \,\!</math> is the matrix vector of variables being solved for; and
*''i'' and ''j'' are cell indices.

== Coefficient matrix ==
The coefficient matrix is:

* an ''N'' x ''N'' square matrix, where ''N'' is the number of cells in the mesh;
* sparse; and
* diagonally dominant.

For example, given a simple 3 x 3 orthogonal mesh (in 2-dimensions):

{| class="wikitable" border="1"
|-
|  -1- || -2- || -3-
|-
|  -6- || -5- || -4-
|-
|  -7- || -8- || -9-
|}

The coefficient matrix might look like:

{| class="wikitable" border="1"
|+ Non-zero coefficients
! i !! -1- !! -2- !! -3- !! -4- !! -5- !! -6- !! -7- !! -8- !! -9-
|- align="center"
! -1-
| X || N || || || || N || || ||
|- align="center"
! -2-
| O || X || N || || N || || || ||
|- align="center"
! -3-
| || O || X || N || || || || ||
|- align="center"
! -4-
| || || O || X || N || || || || N
|- align="center"
! -5-
| || O || || O || X || N || || N ||
|- align="center"
! -6-
| O || || || || O || X || N || ||
|- align="center"
! -7-
| || || || || || O || X || N ||
|- align="center"
! -8-
| || || || || O || || O || X || N
|- align="center"
! -9-
| || || || O || || || || O || X
|}
Where:

* X indicates diagonal cells; and
* O and N indicate non-zero off-diagonal cells.

Since the matrix is sparse, only the non-zero entries need to be stored, something that is achieved by lduAddressing used by the [[matrices in OpenFOAM]].  lduAddressing works by recognizing that:

* Every cell has a diagonal coefficient; and
* Every cell has off-diagonal coefficients for each of their neighbours.

Therefore the diagonal coefficients are stored in an ''N''-long vector array, where ''N'' is the number of cells.  The number of off-diagonal coefficients is equal to the number of cell-pairs that directly influence one another in the linearized equations.  At the matrix level in [[OpenFOAM]], this only includes ''adjacent'' cells.  Therefore the number of off-diagonal coefficients is equal to the number of shared ''faces'' in the mesh.

=== Storage in OpenFOAM ===
[[OpenFOAM]] stores:

* diagonal coefficients as a scalar field, indexed by ''cell volume''; and
* off-diagonal coefficients as two scalar fields, indexed by ''face centres''.

lduAddressing is used to related the two indexing methods.

== Relationship with discretization coefficients ==

Every matrix coefficient has a corresponding [[discretization coefficients|discretization coefficient]].  They are related according to:

<math>A_p = A_{pp}\,\!</math>

<math>A_{r,p} = - \left \lbrack A_{ij} \right \rbrack_{i \neq j}</math>

That is, the off-diagonal matrix coefficients are opposite in sign to their corresponding discretization coefficient; whereas the diagonal has the same sign as its corresponding [[discretization coefficients|discretization coefficient]].

[[Category:Numerical methods]]