Computing the Correction Functions 7,p

At this point we may plow ahead and compute the PDE for each corresponding scheme, but a more efficient route is offered by working with the structure of the inhomogeneity in (7.21). We may make use of the identity 

The remaining two permutations of the possible [XYZ] schemes have a similar relationship, with 

This reduces our workload as we can find the leading order behavior for all six methods by computing the solution to two PDEs. This is not necessarily trivial, but it turns out that the structure of the equations is simple enough that we can find exact solutions. We work now to compute the right-hand side of the governing PDE given in (7.21). 

In order to solve (7.21) for the [ABO scheme, we need to compute the right hand side of (7.22). First, note that if we define the function g(q,p) as 

Now notice that for a function i ( ) purely of position, we have 

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