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.

1 Definition

The matrix coefficients have the form:

$A_{ii} \boldsymbol \psi_i + \sum_j A_{ij} \boldsymbol \psi_j = \mathbf{S}_i$

or:

$\mathbf{A}\boldsymbol \psi = \mathbf S$

Where:

• A are the matrix coefficients;
• $\boldsymbol \psi \,\!$ is the matrix vector of variables being solved for; and
• i and j are cell indices.

2 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):

-1-	-2-	-3-
-6-	-5-	-4-
-7-	-8-	-9-

The coefficient matrix might look like:

Non-zero coefficients

i	-1-	-2-	-3-	-4-	-5-	-6-	-7-	-8-	-9-
-1-	X	N				N
-2-	O	X	N		N
-3-		O	X	N
-4-			O	X	N				N
-5-		O		O	X	N		N
-6-	O				O	X	N
-7-						O	X	N
-8-					O		O	X	N
-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.

2.1 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.

3 Relationship with discretization coefficients

Every matrix coefficient has a corresponding discretization coefficient. They are related according to:

$A_p = A_{pp}\,\!$

$A_{r,p} = - \left \lbrack A_{ij} \right \rbrack_{i \neq j}$

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 coefficient.