The non-existence of symplectic multi-derivative Runge-Kutta methods.

*(English)*Zbl 0816.65043The 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 (Unpublished manuscript)] has studied this property of multi- derivative ($q$) Runge-Kutta methods.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 $(q\le 1)$ the conditions of Lasagni for symplecticness are also necessary, so there are no symplectic multi- derivative Runge-Kutta methods.

Reviewer: I.Coroian (Baia Mare)

##### 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 |

##### Keywords:

non-existence; symplectic multi-derivative Runge-Kutta methods; symplectic methods; irreducible methods; Hamiltonian system##### 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. · Zbl 0451.65063 · 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 Unpublished Manuscript.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 |