  In this paper we consider splitting methods for nonlinear ordinary
differential equations in which one of the (partial) flows that results from
the splitting procedure can not be computed exactly. Instead, we insert a
well-chosen state $y_{\star}$ into the corresponding nonlinearity $b(y)y$,
which results in a linear term $b(y_{\star})y$ whose exact flow can be
determined efficiently. Therefore, in the spirit of splitting methods, it is
still possible for the numerical simulation to satisfy certain properties of
the exact flow. However, Strang splitting is no longer symmetric (even though
it is still a second order method) and thus high order composition methods are
not easily attainable. We will show that an iterated Strang splitting scheme
can be constructed which yields a method that is symmetric up to a given order.
This method can then be used to attain high order composition schemes. We will
illustrate our theoretical results, up to order six, by conducting numerical
experiments for a charged particle in an inhomogeneous electric field, a
post-Newtonian computation in celestial mechanics, and a nonlinear population
model and show that the methods constructed yield superior efficiency as
compared to Strang splitting. For the first example we also perform a
comparison with the standard fourth order Runge--Kutta methods and find
significant gains in efficiency as well better conservation properties.
