3.22 Bounded advection discretisation
Sec. 2.9 described the conservation and boundedness qualities of the advective derivatives and , respectively. According to Eq. (2.31 ), the terms are equivalent when , i.e. when mass conservation Eq. (2.8 ) is satisﬁed in the case when constant.
During computation of an incompressible ﬂow, numerical error can be signiﬁcant, in particularly at the beginning of a steadystate calculation. The governing equations are not satisﬁed, including mass conservation, so .
Discretisation of an advection term of the form as described in Sec. 3.9 can cause unboundedness in . If has a physical bound that is violated, the solution rapidly becomes unstable.
The addition of the term ensures boundedness while , at the expense of conservation. Once , the solution is both bounded and conservative.

(3.43) 

Capturing physics
Boundedness and conservation can be compromised when a term in an equation does not capture the physics of the problem. An example from ﬂame speed combustion modelling uses a parameter which represents the fraction of the unburnt fuel mixture.
The equation for includes the source term , where is a calculated ﬂame speed. Its inclusion could cause the solution of to fall below its lower bound of 0.
Multiplying and dividing by changes the term to , where and unit vector . In this form, the term represents the nonconservative advection of by a ﬂame velocity moving in the direction , which captures the physical nature of combustion. Boundedness can then be maintained by a suitable choice of advection scheme.