In this paper we consider splitting methods for nonlinear ordinarydifferential equations in which one of the (partial) flows that results fromthe splitting procedure can not be computed exactly. Instead, we insert awell-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 bedetermined efficiently. Therefore, in the spirit of splitting methods, it isstill possible for the numerical simulation to satisfy certain properties ofthe exact flow. However, Strang splitting is no longer symmetric (even thoughit is still a second order method) and thus high order composition methods arenot easily attainable. We will show that an iterated Strang splitting schemecan 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 willillustrate our theoretical results, up to order six, by conducting numericalexperiments for a charged particle in an inhomogeneous electric field, apost-Newtonian computation in celestial mechanics, and a nonlinear populationmodel and show that the methods constructed yield superior efficiency ascompared to Strang splitting. For the first example we also perform acomparison with the standard fourth order Runge--Kutta methods and findsignificant gains in efficiency as well better conservation properties.
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