High order splitting methods for analytic semigroups exist

In this paper, we are concerned with the construction and analysis of high order exponential splitting methods for the time integration of abstract evolution equations which are evolved by analytic semigroups. We derive a new class of splitting methods of orders three to fourteen based on complex coefficients. An optimal convergence analysis is presented for the methods when applied to equations on Banach spaces with unbounded vector fields. These results resolve the open question whether there exist splitting schemes with convergence rates greater then two in the context of semigroups. As a concrete application we consider parabolic equations and their dimension splittings. The sharpness of our theoretical error bounds is further illustrated by numerical experiments.