## 4.11Transform condition

Some boundary conditions represent at the boundary as a transformation of the cell value . They can be expressed in terms of a general transform condition

 (4.13)
where is the geometric transformation of variable by a tensor . The transformation is calculated as follows:
 (4.14)
When is a scalar, the transform condition is equivalent to a zero gradient condition. Otherwise, it is implemented so that terms in in Eq. (4.13 ) contribute to coeﬃcients in . These contributions are implicit which improves convergence when solving the matrix equation .

A factor is introduced to specify the contribution to internal coeﬃcients. It represents a single internal coeﬃcient for each component of so has the same rank as , i.e. it is a vector when is a vector, and a tensor when is a tensor.

The multiplication of each coeﬃcient of by its respective component of is denoted by by . In the case of vector , this “component multiplication” is

For advection discretisation, the face value is represented as
 (4.15)
where is an explicit boundary value, calculated from the expression in Eq. (4.13 ) using the current values . The other explicit term uses the current with “” denoting for a vector and for a tensor.

Laplacian discretisation requires the face normal gradient . Combining with Eq. (4.15 ) gives

 (4.16)
where the explicit gradient is calculated by
 (4.17)
The transform condition is summarised within the table below by the value and gradient contributions.