This paper considers computational methods that split a vector field into three components in the case when both the vector field and the split components might be unbounded. We first employ classical Taylor expansion which, after some algebra, results in an expression for a second-order splitting which, strictly speaking, makes sense only for bounded operators. Next, using an alternative approach, we derive an error expression and an error bound in the same setting which are however valid in the presence of unbounded operators. While the paper itself is concerned with second-order splittings using three components, the method of proof in the presence of unboundedness remains valid (although significantly more complicated) in a more general scenario, which will be the subject of a forthcoming paper.

翻译：本文研究一种在向量场及其分裂分量均可能为无界的情况下，将向量场分解为三个分量的计算方法。我们首先采用经典泰勒展开，经过代数推导后得到二阶分裂表达式，严格来说该表达式仅对界算子成立。随后，通过替代方法，我们在相同框架下推导出误差表达式及误差界，这些结果在无界算子存在时依然有效。尽管本文专注于三分量二阶分裂方法，但本文中针对无界情况的证明方法在更一般场景下依然适用（尽管复杂度显著增加），这将成为后续论文的研究主题。