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 (Unpublished manuscript)] has studied this property of multi- derivative ($q$) Runge-Kutta methods.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

65L06Multistep, Runge-Kutta, and extrapolation methods
65L05Initial value problems for ODE (numerical methods)
37-99Dynamic systems and ergodic theory (MSC2000)
37J99Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems
70H15Canonical and symplectic transformations in particle mechanics
34A34Nonlinear ODE and systems, general
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).
[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. · Zbl 0451.65063 · 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 Unpublished Manuscript.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