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

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 error bounds for two families of second-order exponential splittings. The role of quadratures when integrating the twice-iterated Duhamel's formula is reformulated: we show that their choice defines the structure of the splitting. Furthermore, the reformulation allows us to consider quadratures based on the Birkhoff interpolation to obtain not only pure-stages splittings but also those containing derivatives of $B(t)$ and commutators of $A$ and $B(t)$. In this approach, the construction and error analysis of the splittings are carried out simultaneously. We discuss the accuracy of the members of the families. Numerical experiments are presented to complement the theoretical consideration.