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：我们针对含两个线性分量的线性发展方程，提出斯特朗型指数分裂族$\mathcal{F}(h,\tau)$的一种新颖且直接的推导方法。其中一个分量假定为时间无关的无界算子，另一分量为显式时间依赖的有界算子。该族$\mathcal{F}(h,\tau)$由步长$h$和连续参数$\tau$共同表征，其中$\tau$定义了族中的每个成员。研究表明，推导过程与误差分析仅需两个基础论证：常数变易公式和单形区域上的特定求积公式。针对这些斯特朗型分裂方法，我们证明了其收敛性——根据相关算子对易子的性质，该收敛可为一阶或二阶。误差界以对易子界的形式呈现。基于误差项的显式表达，我们确立了$\tau$对$\mathcal{F}(h,\tau)$精度的影响规律，从而可探究最优参数$\tau$。这种简洁而高效的方法建立了指数积分器与分裂方法间的关联。此外，本方法可轻松推广至相似框架下高阶分裂方法的推导。毋庸置疑，所得结果同样适用于时间无关算子情况下的斯特朗型分裂。为补充理论结果，我们基于线性薛定谔方程展示了不同$\tau$取值下的数值实验。