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There are lots of higher order time splitting method as shown by the list with real and complex coefficients $a_i, b_i, c_i$:

$$ [e^{c_s \Delta t \hat C}] e^{b_s \Delta t \hat B} e^{a_s \Delta t \hat A} ... [e^{c_1 \Delta t \hat C}] e^{b_1 \Delta t \hat B} e^{a_1 \Delta t \hat A} u $$

It is not clear which one should I choose, so are there any advantage of complex coefficients over real coefficients? Also, which method is better in practices. Intuitively, the least step $s$ and real coefficients seems an easy choice.

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The design of the "best" splitting schemes is discussed and investigated at length in this recent paper (disclosure: I am one of its authors). In short, the most commonly used criterion is the size of the leading truncation error term coefficients. This can generally be made smaller by using a larger number of stages, and the tradeoff can be worthwhile if your step size is limited by accuracy considerations in practice.

I don't know of any inherent advantage to methods with complex coefficients; generally that is a disadvantage since you have to do complex arithmetic which requires more operations.

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  • $\begingroup$ I just notice that you made the website which is very useful. Thanks. Focus only on the LEM, then the "best" 2nd and 4th order scheme are "AM 3-2" and "Emb 4/3 BM PRK/A" respectively. Am I understand it correctly? I remember that negative coefficient cannot be used when the dynamic equation is not time reversal symmetry. But the real part of all complex scheme are positive, does it means that they can be used in such situation? Hope that you can clarify a bit in your answer. $\endgroup$ – unsym Jul 30 '16 at 18:35
  • $\begingroup$ I am actually looking for a replacement for the adaptive RK45 which is bad because of the nature of my problem. So the "Emb 5/4 A" can serve similar purpose? $\endgroup$ – unsym Jul 30 '16 at 18:38

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