An equation may include a function which can return a value which is positive or negative, e.g.
The solution variable could have a physical bound at any low value and/or high value . We can treat the function in Eq. (3.37 ) as follows to maintain boundedness:
If we examine the situation with , Eq. (3.37 ) would be discretised as
An equivalent analysis shows Eq. (3.40 ) maintains boundedness for the upper bound . In that case, note that is negative so the term in can still be treated implicitly.
Many properties are expressed as a fraction, e.g. concentration of a chemical species, so are bounded by 0 and 1. When that happens, Eq. (3.39 ) simpliﬁes to
It is particularly important to maintain boundedness of a ﬁeld with 0-1 bounds when it is used to calculate another property which has a physical bound. For example, in a system of ﬂuid phases, ﬂuid can be calculated by from phase fractions and densities .
A small amount of unboundedness in , in the phase with highest , causes large unboundedness in , e.g. in 2 phases with and , the calculated with .