We present a derivation and error bound for the family of fourth order splittings, originally introduced by Chin and Chen, where one of the operators is unbounded and the second one bounded but time dependent, and which are dependent on a parameter. We first express the error by an iterated application of the Duhamel principle, followed by quadratures of Birkhoff-Hermite type of the underlying multivariate integrals. This leads to error estimates and bounds, derived using Peano/Sard kernels and direct estimates of the leading error term. Our analysis demonstrates that, although no single value of the parameter can minimise simultaneously all error components, an excellent compromise is the cubic Gauss--Legendre point $1/2-\sqrt{15}/10$.

翻译：本文针对Chin和Chen最初提出的四阶分裂法族，给出了其推导过程与误差界，其中涉及一个无界算子和一个有界但随时间变化的算子，且该方法依赖于一个参数。我们首先通过Duhamel原理的迭代应用来表达误差，随后对相关的多元积分进行Birkhoff-Hermite型求积。利用Peano/Sard核函数以及对主导误差项的直接估计，我们推导出了误差估计与误差界。分析表明，尽管不存在单一参数值能够同时最小化所有误差分量，但立方Gauss–Legendre点$1/2-\sqrt{15}/10$是一个极佳的折中选择。