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 (Unpublished manuscript)] has studied this property of multi- derivative (\(q\)) Runge-Kutta methods.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

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.

Reviewer: I.Coroian (Baia Mare)

### MSC:

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 |

### Keywords:

non-existence; symplectic multi-derivative Runge-Kutta methods; symplectic methods; irreducible methods; Hamiltonian system
Full Text:
DOI

### 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). · 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 |

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