Second Order Exponential Splittings in the Presence of Unbounded and Time-Dependent Operators: Construction and Convergence

SIAM Journal on Numerical Analysis, Volume 63, Issue 1, Page 288-305, February 2025.
Abstract. For linear differential equations of the form [math], [math], with a possibly unbounded operator [math], we construct and deduce error bounds for two families of second-order exponential splittings. The role of quadratures when integrating the twice-iterated Duhamel’s formula is reformulated: we show that their choice defines the structure of the splitting. Furthermore, the reformulation allows us to consider quadratures based on the Birkhoff interpolation to obtain splittings featuring not only exponentials of [math] or [math] but also time-derivatives of [math] and commutators of [math] and [math]. In this approach, the construction and error analysis of the splittings are carried out simultaneously. We discuss the accuracy of the members of the families. Numerical experiments are presented to complement the theoretical consideration.