# OpenFOAM guide/FvMatrix;;H

An implementation of the H operator, an off-diagonal function used in linearizing the convection term. The momentum equation can be expressed as:

$A_p \mathbf{U}_p = \mathbf{H}(\mathbf{U}) - \Delta p$

Where:

• The subscript p is the cell volume index;
• A are discretization coefficients;
• U is velocity;
• p is pressure; and
• H is given by:

By face index:

$\mathbf{H}_{o,i} = A_{n,i} \boldsymbol \psi_{n,i} + \mathbf{S}_i$

$\mathbf{H}_{n,i} = A_{o,i} \boldsymbol \psi_{o,i} + \mathbf{S}_i$

Where:

• A are matrix coefficients;
• o and n are the owner and neighbour indices;
• i is the face index;
• $\boldsymbol\psi$ is the fvMatrix field variable (usually velocity, 'U);
• S is the discretization source term.

By cell volume index:

$\mathbf{H}_p = \sum\limits_r A_r \boldsymbol \psi_r + \mathbf{S}_p$

Where:

• r are related cells[1].

The calculation of the sum of related cells is performed by lduMatrix;;H.

## Notes

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.