Order conditions for integrators and mappings with applications Original Research Article

We regard the elementary weight function as a mapping from input to output, so that the mappings allow us to derive the order conditions for a given method more easily and concisely. This paper presents the order conditions of one-step Runge–Kutta methods, Runge–Kutta-type methods with derivatives, a class of two-step Runge–Kutta methods and three-step Runge–Kutta methods, based on the elementary weight mappings, the composition formula of two mappings as well as the mappings I,Dr,E(ν)I,Dr,E(ν) defined on the set TT of all rooted trees from an input to an output. It is pointed out that the order conditions can be easily achieved for linear multistep methods, and second-derivative linear multistep methods of order pp, and it suffices to consider only bushy trees of order up to pp together with the empty tree 0̸0̸ for a linear multistep method, or a second-derivative linear multistep method of order pp