An elementary approach to splittings of unbounded operators
Arieh Iserles, Karolina Kropielnicka
TL;DR
This paper develops an elementary, semigroup-based analysis of exponential splittings for unbounded operators $A$ and $B$ generating $e^{tA}$, $e^{tB}$, and $e^{t(A+B)}$. By applying the Duhamel principle, it derives explicit integral expressions for the errors of Lie--Trotter, Palindromic Lie--Trotter, and Strang splittings, establishing that LT is first-order, while PLT and Strang achieve second- and third-order accuracy, respectively. The authors then convert these expressions into tight operator-norm bounds using logarithmic norms, introducing $\omega = \mu[A]+\mu[B]-\mu[A+B]$ and bounding errors in terms of commutators $[A,B]$, $[A,[A,B]]$, and $[B,[A,B]]$, with simplifications when the generators have zero logarithmic norm. The results provide practical, implementable guidelines for choosing splittings in numerical simulations of evolution equations and demonstrate that an accessible, undergraduate-friendly approach can yield rigorous error guarantees for unbounded operators.
Abstract
Using elementary means, we derive the three most popular splittings of $e^{(A+B)}$ and their error bounds in the case when $A$ and $B$ are (possibly unbounded) operators in a Hilbert space, generating strongly continuous semigroups, $e^{tA}$, $e^{tB}$ and $e^{t(A+B)}$. The error of these splittings is bounded in terms of the norm of the commutators $[A, B]$, $[A, [A, B]]$ and $[B, [A, B]]$.