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In this paper, we study symmetric integrators for solving second-orderordinary differential equations on the basis of the notion of continuous-stageRunge-Kutta-Nystrom methods. The construction of such methods heavily relies onthe Legendre expansion technique in conjunction with the symmetric conditionsand simplifying assumptions for order conditions. New families of symmetricintegrators as illustrative examples are presented. For comparing the numericalbehaviors of the presented methods, some numerical experiments are alsoreported.

Symmetric integrators based on continuous-stage Runge-Kutta-Nystrom methods for reversible systems