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The non-existence of symplectic multi-derivative Runge-Kutta methods. (English) Zbl 0816.65043

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 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 (Unpublished manuscript)] has studied this property of multi- derivative ($q$) Runge-Kutta methods.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:

1) It is shown that an irreducible Runge-Kutta method can be symplectic only for $q\le 1$, i.e., for standard Runge-Kutta methods.

2) It is shown that in this case $\left(q\le 1\right)$ the conditions of Lasagni for symplecticness are also necessary, so there are no symplectic multi- derivative Runge-Kutta methods.

##### MSC:
 65L06 Multistep, Runge-Kutta, and extrapolation methods 65L05 Initial value problems for ODE (numerical methods) 37-99 Dynamic systems and ergodic theory (MSC2000) 37J99 Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems 70H15 Canonical and symplectic transformations in particle mechanics 34A34 Nonlinear ODE and systems, general
##### References:
 [1] M. P. Calvo and J. M. Sanz-Serna,Canonical B-series. To appear in Numer. Math. [2] G. Dahlquist and R. Jeltsch,Generalized disks of contractivity for explicit and implicit Runge-Kutta methods. TRITA-NA Report 7906 (1979). [3] E. Hairer,Backward analysis of numerical and symplectic methods. To appear in Advances in Comput. Math. [4] E. Hairer, S. P. Nørsett and G. Wanner,Solving Ordinary Differential Equations I. Nonstiff Problems. Second Edition. Springer-Verlag, Berlin, 1993. [5] W. H. Hundsdorfer and M. N. Spijker,A note on B-stability of Runge-Kutta methods. Numer. Math. 36(1981), p. 319–331. · doi:10.1007/BF01396658 [6] F. M. Lasagni,Integration methods for Hamiltonian differential equations. Unpublished Manuscript. [7] J. M. Sanz-Serna,Symplectic integrators for Hamiltonian problems: an overview. Acta Numerica 1 (1992), p. 243–286. · doi:10.1017/S0962492900002282