Fourth-order compact exponential splittings for unbounded operators

TL;DR

This paper derives error bounds for a family of fourth-order exponential splitting methods involving unbounded operators, identifying an optimal parameter choice for minimizing errors.

Contribution

It provides a detailed error analysis and bounds for Chin and Chen's fourth-order splittings with unbounded and time-dependent operators, including an optimal parameter selection.

Findings

Error bounds derived using Duhamel principle and quadratures.

Optimal parameter identified at cubic Gauss--Legendre point.

Analysis shows no single parameter minimizes all error components.

Abstract

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 $1/2-\sqrt{15}/10$.