Perturbative Expansion Exponentially Small Errors

Since Hh is assumed to be constant along the numerical solution, the coordinate transformations have the result of giving an effective order of four for the energy. It turns out that the Takahashi-Imada method is, more generally, an effective 4th order scheme, i.e. for arbitrary quantities, not just the energy [166], 

Let us summarize the general situation with regard to backward error analysis. Assume a smooth differential equation system 

Now in the case of a Hamiltonian system (with Hamiltonian H) solved using a symplectic numerical method, the above construction can generally be carried out to any desired order by matching of terms in Taylor expansions, assuming sufficient differentiability. In this case it will turn out that 

We have worked out formulas above for the first terms in the expansion in the case of several splitting methods. We can thus view (at least locally) our numerical trajectory as the stroboscopic map (snapshots) of the evolution of a continuous Hamiltonian system. 

Ideally, we would like to be able write that the solution to the modified equations is equivalent to mapping via the specified one-sfep method  

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