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.

翻译：本文针对含两个线性分量的一般线性演化方程，提出了一族Strange型指数分裂$\mathcal{F}(h,\tau)$的新颖简洁推导方法。其中一分量假定为时间独立无界算子，另一分量则为具有显式时间依赖性的有界算子。该族$\mathcal{F}(h,\tau)$由时间步长$h$和定义族成员的连续参数$\tau$共同刻画。研究表明，其推导与误差分析仅需借助两个基本论证：常数变易公式及单纯形积分上的特定求积法则。针对此类Strange型分裂，我们证明了其收敛性——根据相关算子交换子的性质，收敛阶可为一阶或二阶。据此，误差界可通过交换子边界表示。基于误差项的显式形式，我们确立了$\tau$对$\mathcal{F}(h,\tau)$精度的作用机制，进而得以研究$\tau$的最优取值。这种简洁而强大的方法建立了指数积分器与分裂方法之间的内在联系。此外，本方法可轻松推广至更高阶分裂方法在类似条件下的推导。不言而喻，所得结论同样适用于时间无关算子情形下的Strange型分裂。为补充严密理论结果，我们基于线性薛定谔方程开展了不同$\tau$值下的数值实验。