Fourth-order compact exponential splittings for unbounded operators

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