## Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential EquationsNumerical methods that preserve properties of Hamiltonian systems, reversible systems, differential equations on manifolds and problems with highly oscillatory solutions are the subject of this book. A complete self-contained theory of symplectic and symmetric methods, which include Runge-Kutta, composition, splitting, multistep and various specially designed integrators, is presented and their construction and practical merits are discussed. The long-time behaviour of the numerical solutions is studied using a backward error analysis (modified equations) combined with KAM theory. The book is illustrated by many figures, it treats applications from physics and astronomy and contains many numerical experiments and comparisons of different approaches. |

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### Contents

Numerical Integrators | 23 |

6 | 46 |

Order Conditions Trees and BSeries | 47 |

XI | 55 |

Conservation of First Integrals and Methods on Manifolds | 93 |

Symmetric Integration and Reversibility | 131 |

V6 Exercises | 164 |

Further Topics in Structure Preservation | 209 |

Backward Error Analysis and Structure Preservation | 287 |

Hamiltonian Perturbation Theory and Symplectic Integrators | 327 |

Reversible Perturbation Theory and Symmetric Integrators | 375 |

Dissipatively Perturbed Hamiltonian and Reversible Systems | 391 |

Highly Oscillatory Differential Equations | 407 |

Dynamics of Multistep Methods | 455 |

492 | |

509 | |

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### Common terms and phrases

action-angle variables adjoint Algorithm approximation B-series coefficients collocation methods composition methods computation conservation consider constant corresponding defined Definition denote derivative differential equation equivalent Euler method exact solution Example exponentially formula function Gauss methods given Hamiltonian system identity implicit midpoint rule implies initial values invertible iteration Jacobi identity Kepler problem Lemma Lie algebra Lie derivative Lie group long-time behaviour manifold matrix method of order method ph modified equation multistep methods nonlinear numerical solution Nyström methods obtained one-step method order conditions order methods orthogonal p-reversible parametrization partitioned Runge-Kutta methods perturbation ph(y Poisson bracket polynomial projection proof of Theorem proves result reversible satisfies Sect shows skew-symmetric splitting methods step size h Störmer/Verlet method Störmer/Verlet scheme Switching Lemma symmetric matrix symmetric methods symplectic Euler method symplectic integrator symplectic method symplectic transformation tangent space tion torus trapezoidal rule trees truncated vector field yields