On the necessity of negative coefficients for operator splitting schemes of order higher than two

In this paper we analyse numerical integration methods applied to differential equations which are separable in solvable parts. These methods are compositions of flows associated with each part of the system. We propose an elementary proof of the necessary existence of negative coefficients if the schemes are of order, or effective order, p⩾3 and provide additional information about the distribution of these negative coefficients. It is shown that if the methods involve flows associated with more general terms this result does not necessarily apply and in some cases it is possible to build higher order schemes with positive coefficients.