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**The midpoint scheme and variants for Hamiltonian systems: Advantages and pitfalls.**
*(English)*
Zbl 0947.65133

The authors study the suitability of the (implicit) midpoint rule for the integration of highly oscillatory Hamiltonian systems as they often appear in molecular dynamics simulations. They show that one must impose rather tight restrictions on the stepsize in order to be sure that no misleading results are obtained. In particular, they show that even the errors in slowly varying quantities (adiabatic invariants) like the energy may exhibit an unexpected growth, if a too large stepsize is used. To make things worse, the numerical results may give no indications that something goes wrong. The analysis is based on the detailed examination of some simple examples (mainly stiff oscillators, single and coupled). All theoretical results are tested in numerical computations.

Reviewer: Werner M.Seiler (Mannheim)

### MSC:

65P10 | Numerical methods for Hamiltonian systems including symplectic integrators |

65L20 | Stability and convergence of numerical methods for ordinary differential equations |

37J45 | Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010) |

65L06 | Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations |

37M15 | Discretization methods and integrators (symplectic, variational, geometric, etc.) for dynamical systems |

65L50 | Mesh generation, refinement, and adaptive methods for ordinary differential equations |