Family of Strang-type exponential splittings in the presence of unbounded and time dependent operators

We present a new and straightforward derivation of a family $\mathcal{F}(h,\tau)$ of exponential splittings of Strang-type for the general linear evolutionary equation with two linear components. One component is assumed to be a time-independent, unbounded operator, while the other is a bounded one with explicit time dependence. The family $\mathcal{F}(h,\tau)$ is characterized by the length of the time-step $h$ and a continuous parameter $\tau$, which defines each member of the family. It is shown that the derivation and error analysis follows from two elementary arguments: the variation of constants formula and specific quadratures for integrals over simplices. For these Strang-type splittings, we prove their convergence which, depending on some commutators of the relevant operators, may be of first or second order. As a result, error bounds appear in terms of commutator bounds. Based on the explicit form of the error terms, we establish the influence of $\tau$ on the accuracy of $\mathcal{F}(h,\tau)$, allowing us to investigate the optimal value of $\tau$. This simple yet powerful approach establishes the connection between exponential integrators and splitting methods. Furthermore, the present approach can be easily applied to the derivation of higher-order splitting methods under similar considerations. Needless to say, the obtained results also apply to Strang-type splittings in the case of time independent-operators. To complement rigorous results, we present numerical experiments with various values of $\tau$ based on the linear Schr\"odinger equation.