3.20 Other terms
In an equation for , there can be terms other than the time derivative, advection and Laplacian of , including:
- a linear function where is a scalar coeﬃcient or ﬁeld;
- a term without , sometimes including the derivative of another variable, e.g. .
The second example is simply discretised as an explicit gradient term as described in Sec. 3.15 . Derivative terms like this one are described in earlier sections and require no further discussion.
The ﬁrst example, the term, requires much more discussion, particularly relating to the possibility of implicit discretisation. Let us consider the following equation:
Therefore, such a term must be discretised explicitly when it has a negative sign (or positive on the right side of “=”), to ensure the matrix equation is solvable. The nature of Eq. (3.33 ) is that can only increase from an initial positive value.
Implicit discretisation of linear terms
Let us now consider the equivalent equation with a linear term that has a positive sign, i.e.
It is important to maintain a physical bound of . Discretisation of Eq. (3.34 ) using the Euler time scheme Eq. (3.21 ) gives
Even when the term is not linear in , it can be implemented as such by “dividing and multiplying by ”. For example, the turbulence model includes an term in the equation, i.e. , ignoring other terms. Dividing and multiplying the term by gives