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Backward analysis of numerical integrators and symplectic methods

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E. Hairer

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Abstract. A backward analysis of integration methods, whose numerical solution is a P-series, is presented. Such methods include Runge-Kutta methods, partitioned Runge-Kutta methods and Nystr\"om methods. It is shown that the numerical solution can formally be interpreted as the exact solution of a perturbed differential system whose right-hand side is again a P-series. The main result of this article is that for symplectic integrators applied to Hamiltonian systems the perturbed differential equation is a Hamiltonian system too. The proofs use the one-to-one correspondence between rooted trees and the expressions appearing in the Taylor expansions of the exact and numerical solutions (elementary differentials).

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Key Words. Backward analysis, Hamiltonian systems, Runge-Kutta methods, symplectic methods, P-series.

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