Error bounds for splitting methods in unitary problems

TL;DR

This paper provides a systematic analysis of local and global errors in splitting methods for unitary problems, with explicit bounds and error estimates applicable to various schemes.

Contribution

It introduces new error bounds for splitting methods in unitary problems, including operator norm and commutator-based estimates, especially for two-operator cases.

Findings

Derived explicit error bounds for splitting schemes

Presented operator norm and commutator-based error estimates

Illustrated results with specific scheme examples

Abstract

Splitting methods constitute a widely used class of numerical integrators for ordinary and partial differential equations, particularly well suited to problems that can be decomposed into simpler subproblems. High-order splitting schemes are available that achieve high accuracy while preserving key qualitative properties of the underlying dynamical system, and are successfully used across a broad range of fields. In this work, we present a systematic analysis of both local and global errors arising from arbitrary splitting methods applied to unitary problems. Two complementary types of error estimates are derived. The first is expressed in terms of operator norms, while the second is formulated using norms of commutators and can, under suitable assumptions, be extended to certain classes of unbounded operators. Special attention is devoted to the case where only two operators are involved. The theoretical results are illustrated by deriving explicit error bounds for some representative schemes.