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Some Properties of Symplectic Runge-Kutta Methods

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E. Hairer and P. Leone

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Abstract. We prove that to every rational function $R(z)$ satisfying $R(-z)R(z)=1$, there exists a symplectic Runge-Kutta method with $R(z)$ as stability function. Moreover, we give a surprising relation between the poles of $R(z)$ and the weights of the quadrature formula associated with a symplectic Runge-Kutta method.

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Key Words. Symplectic Runge-Kutta methods, $W$-transformation, poles of stability function, weights of quadrature formula

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