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A new approach to the algebraic structures for integration methods. (English) Zbl 1016.65047

Summary: The analysis of compositions of Runge-Kutta methods involves manipulations of functions defined on rooted trees. Existing formulations due to J. C. Butcher [Math. Comput. 26, 79-106 (1972; Zbl 0258.65070)], E. Hairer and G. Wanner [Computing 13, 1-15 (1974; Zbl 0293.65050)], and A. Murua and J. M. Sanz-Serna [Philos. Trans. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 357, No. 1754, 1079-1100 (1999; Zbl 0946.65056)], while equivalent, differ in details.
The subject of the present paper is a new recursive formulation of the composition rules. This both simplifies and extends the existing approaches. Instead of using the error conditions based on trees, we propose the construction of the order conditions using a suitably chosen basis on the tree space. In particular, the linear structure of the tree space gives a representation of the \(C\) and \(D\) simplifying assumptions on trees which is not restricted to Runge-Kutta methods. A proof of the group structure of the set of elementary weight functions satisfying the \(D\) simplifying assumptions is also given is this paper.

MSC:

65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65L05 Numerical methods for initial value problems involving ordinary differential equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
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