21.2.1 Rules for manipulating operators

In general, finite difference derivative and average operators don't commute unless the grid resolution is constant. Assuming that $\alpha_{i}$ is defined at grid points within T-cells, then the above condition is illustrated by the following

$$\delx(\overline{\alpha_{i}}^\lambda) \ne \overline{\delx(\alpha_{i})}^\lambda$$ (21.9)

How is a term like $\delx(\overline{\alpha_{i}}^\lambda)$ evaluated? It can be expanded from the inside out as

$$\delx(\overline{\alpha_{i}}^\lambda) \delx(\frac{\alpha_{i} + \alpha_{i+1}}{2})$$ =

$$\frac{(\alpha_{i+1} + \alpha_{i+2})/2 - (\alpha_{i} + \alpha_{i+1})/2}{dxt_{i+1}}$$ =

$$\frac{\alpha_{i+2} - \alpha_{i}}{2\cdot dxt_{i+1}}$$ = (21.10)

or from the outside in as

 

$$\delx(\overline{\alpha_{i}}^\lambda) \frac{\overline{\alpha_{i+1}}^\lambda - \overline{\alpha_{i}}^\lambda}{dxt_{i+1}}$$ =

$$\frac{(\alpha_{i+1} + \alpha_{i+2})/2 - (\alpha_{i} + \alpha_{i+1})/2}{dxt_{i+1}}$$ =

$$\frac{\alpha_{i+2} - \alpha_{i}}{2\cdot dxt_{i+1}}$$ = (21.11)

Both results are equal. Now expand the following:

 

$$\overline{\delx(\alpha_{i})}^\lambda \frac{(\alpha_{i+1}-\alpha_{i})/dxu_i + (\alpha_{i+2}-\alpha_{i+1})/dxu_{i+1} }{2}$$ = (21.12)

Equation (21.12) is only equal to Equation (21.11) when

dxt_i+1 = dxu_i = dxu_i+1 (21.13)

Also, it is worth remembering that the results of operators are displaced by the distance of a half cell width. For example, the single operator $\delx(\alpha_{i,k,j})$ results in a quantity defined on the eastern face^21.1 of cell and the double operator $\delx(\delx(\alpha_{i,k,j}))$ results in a quantity defined at the grid point within T_i+1,k,j. These results easily extend to two and three dimensions. Mixed double operators such as $\delx(\overline{\alpha_{i,k,j}}^\phi)$ results in a quantity defined on the grid point within cell .