Fourth-order compact exponential splittings for unbounded operators

We present a formal proof of order and an error bound for the family of fourth-order compact splittings, where one of the operators is unbounded and the second one bounded but time-dependent, and which depends on a parameter. We first express the error by an iterated application of the Duhamel’s principle, followed by quadratures of Birkhoff–Hermite type of 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 − √15/10.