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 steady-state 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) This additional term is implicit in , so contributes to the diagonal coeﬃcients of the matrix. If is positive, we might expect a singular matrix, as discussed in Sec. 3.20 . However, the decrease to the diagonal coeﬃcient is matched by an increase from the discretisation of . If that term uses the upwind scheme, the discretisation can be split into a contribution from positive outgoing ﬂuxes and negative incoming ﬂuxes . In extensive form (i.e. scaled by , see Sec. 3.24 ), using values in the cell of interest and neighbouring cells , the terms are The terms cancel, leaving negative coeﬃcients for neighbouring cells (since is negative). The diagonal coeﬃcient is then equal to the sum of magnitude of those neighbour cell coeﬃcients, resulting in an invertible matrix.

### 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 non-conservative 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.

Notes on CFD: General Principles - 3.22 Bounded advection discretisation 