The initial and final half steps, and subsequent interweaving of the position and momentum updates, ensure a very delicate cancellation that yields much smaller local errors and, perhaps more importantly, small global errors even after long integration times.
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You can see this theoretically by deriving the errors using Baker-Campbell-Hausdorff. For example, the first three equations in Section 2 of https://arxiv.org/pdf/1502.01510.pdf ….
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You can also see this empirically but looking at the error in the Hamiltonian after each intermediate step. You get large errors that oscillate up and down until that final half step when there's a magical cancellation.
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