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Zbl 0816.65043
Hairer, E.; Murua, A.; Sanz-Serna, J.M.
The non-existence of symplectic multi-derivative Runge-Kutta methods. (English)
[J] BIT 34, No.1, 80-87 (1994). ISSN 0006-3835; ISSN 1572-9125/e

The authors investigate the numerical solution of a Hamiltonian system, especially the property called``symplecticness'' of a numerical method which consists in the preservation of some differential 2-form. First {\it F. M. Lasagni} [Integration methods for Hamiltonian differential equations. (Unpublished manuscript)] has studied this property of multi- derivative ($q$) Runge-Kutta methods. The main results of this paper are:\par 1) It is shown that an irreducible Runge-Kutta method can be symplectic only for $q \leq 1$, i.e., for standard Runge-Kutta methods.\par 2) It is shown that in this case $(q\leq 1)$ the conditions of Lasagni for symplecticness are also necessary, so there are no symplectic multi- derivative Runge-Kutta methods.
[I.Coroian (Baia Mare)]

MSC 2000:: *65L06 Multistep, Runge-Kutta, and extrapolation methods
65L05 Initial value problems for ODE (numerical methods)
37-99 Dynamic systems and ergodic theory
37J99 Finite-dimensional Hamiltonian etc. systems
70H15 Canonical transformations
34A34 Nonlinear ODE and systems, general

Keywords: non-existence; symplectic multi-derivative Runge-Kutta methods; symplectic methods; irreducible methods; Hamiltonian system

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