Calculation of the Inhomogeneous Term

The computation of the right hand side in (7.28) is required in order to solve the corresponding partial differential equation and obtain the leading order correction function for our symmetric scheme XYZYX]. From the form of the operator in Proposition 7.1, we can see that our life is made considerably easier if we initially consider methods OYZYO], which substitutes 

The inhomogeneity for the schemes OABAO] and OBABO] is hence a combination of the two commutator terms 

We can see immediately that the Fredholm alternative holds in both cases, as both equations have right-hand sides that are of odd order in p, and hence will average to 0. 

Lemma 7.1 Consider two numerical schemes with associated operators governing the evolution of measure ST andTS, with unique associated invariant distributions PsT and pts respectively, such that 

Under suitable ergodicity assumptions (such that in the long-time limit, all distributions converge to the invariant distribution), we have 

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