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 (Unpublished manuscript)] has studied this property of multi- derivative (\(q\)) Runge-Kutta methods.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 \(q \leq 1\), i.e., for standard Runge-Kutta methods.
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.


65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65L05 Numerical methods for initial value problems involving ordinary differential equations
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
70H15 Canonical and symplectic transformations for problems in Hamiltonian and Lagrangian mechanics
34A34 Nonlinear ordinary differential equations and systems
Full Text: DOI


[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). · Zbl 1106.65062
[3] E. Hairer,Backward analysis of numerical and symplectic methods. To appear in Advances in Comput. Math. · Zbl 0828.65097
[4] E. Hairer, S. P. Nørsett and G. Wanner,Solving Ordinary Differential Equations I. Nonstiff Problems. Second Edition. Springer-Verlag, Berlin, 1993. · Zbl 0789.65048
[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 Unpublished Manuscript. · Zbl 0675.34010
[7] J. M. Sanz-Serna,Symplectic integrators for Hamiltonian problems: an overview. Acta Numerica 1 (1992), p. 243–286. · Zbl 0762.65043 · doi:10.1017/S0962492900002282
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.