4.11 Transform 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.