Order conditions for integrators and mappings with applications

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) defined on the set T 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 p, and it suffices to consider only bushy trees of order up to p together with the empty tree 0̸ for a linear multistep method, or a second-derivative linear multistep method of order p.