Splitting methods for unbounded operators

TL;DR

This paper develops and analyzes splitting methods for unbounded operators, providing error bounds and extending classical techniques to more general unbounded cases, with implications for future research.

Contribution

It introduces a novel error analysis for second-order splitting methods involving unbounded operators, extending classical approaches to more general unbounded cases.

Findings

Derived an error expression valid for unbounded operators

Provided an error bound for second-order splitting methods

Extended classical splitting techniques to unbounded operators

Abstract

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.