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For linear differential equations of the form $u'(t)=[A + B(t)] u(t)$,$t\geq0$, with a possibly unbounded operator $A$, we construct and deduce errorbounds for two families of second-order exponential splittings. The role ofquadratures when integrating the twice-iterated Duhamel's formula isreformulated: we show that their choice defines the structure of the splitting.Furthermore, the reformulation allows us to consider quadratures based on theBirkhoff interpolation to obtain not only pure-stages splittings but also thosecontaining derivatives of $B(t)$ and commutators of $A$ and $B(t)$. In thisapproach, the construction and error analysis of the splittings are carried outsimultaneously. We discuss the accuracy of the members of the families.Numerical experiments are presented to complement the theoreticalconsideration.

Second-order exponential splittings in the presence of unbounded and time-dependent operators: construction and convergence