Stability of Exponential Operator Splitting Methods for Noncontractive Semigroups

Operator splitting methods constitute an attractive class of time discretization schemes for evolution equations. Due to their computational advantages, they are widely used in scientific computation. Here we are interested in linear parabolic problems where the evolution is given by an analytic semigroup. Apart from the simple situation where the single operators generate semigroups of contractions, the stability of operator splitting methods in Banach spaces can be a delicate problem. In this paper we derive a Banach space framework that allows us to prove the stability of the Lie--Trotter splitting for noncontractive semigroups under reasonable assumptions on the involved operators. It is noteworthy that the same assumptions also imply stability of the Strang--Marchuk splitting. The stability result is verified with the help of a particular representation of the local error combined with an appropriate commutator bound. Our abstract stability result can be applied, in particular, to the dimension split...