3.20 Other terms

In an equation for eqn, there can be terms other than the time derivative, advection and Laplacian of eqn, including:

  • a linear function eqn where eqn is a scalar coefficient or field;
  • a term without eqn, sometimes including the derivative of another variable, e.g. eqn.

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 eqn term, requires much more discussion, particularly relating to the possibility of implicit discretisation. Let us consider the following equation:

@ ---- c = 0: @t \relax \special {t4ht=

If eqn is discretised implicitly, then the matrix would contain a zero diagonal coefficient when eqn, making it singular or non-invertible. If this occurs, eqn is not present in the linear equation for the relevant cell, so it cannot be solved.

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 eqn 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.

@ ----+ c = 0: @t \relax \special {t4ht=
With Eq. (3.34 ), eqn can only decrease from an initial positive value but reaches a lower limit at eqn. Like many scalar properties, e.g. eqn, eqn, eqn, etc., eqn may have a lower physical bound of 0, which the equation intentionally reflects.

It is important to maintain a physical bound of eqn. Discretisation of Eq. (3.34 ) using the Euler time scheme Eq. (3.21 ) gives

= ---1--- - o; 1+ c t \relax \special {t4ht=
which ensures boundedness since eqn. An equivalent explicit discretisation gives eqn which is only bounded when eqn, similar to the eqn limit imposed by explicit discretisation of advection, described in Sec. 3.17 . To avoid this eqn limit, we apply implicit discretisation to terms with a positive sign on the left side of “=”, where possible.

Even when the term is not linear in eqn, it can be implemented as such by “dividing and multiplying by eqn”. For example, the k turbulence model includes an term in the eqn equation, i.e. @k=@t + , ignoring other terms. Dividing and multiplying the term by eqn gives

producing a linear term in eqn which can be discretised implicitly using a coefficient c = .
Notes on CFD: General Principles - 3.20 Other terms