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 (Unpublished manuscript)] has studied this property of multi- derivative (\(q\)) Runge-Kutta methods.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

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
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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
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[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 Unpublished Manuscript.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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