I.Examples and Numerical Experiments I.1 First Problems and Methods I.1.1 The Lotka-Volterra Model I.1.2 First Numerical Methods I.1.3 The Pendulum as a Hamiltonian System I.1.4 The St6rmer-Verlet Scheme I.2 The Kepler Problem and the Outer Solar System I.2.1 Angular Momentum and Kepler's Second Law I.2.2 Exact Integration of the Kepler Problem I.2.3 Numerical Integration of the Kepler Problem I.2.4 The Outer Solar System I.3 The Henon-Heiles Model I.4 Molecular Dynamics I.5 Highly Oscillatory Problems I.5.1 A Fermi-Pasta-Ulam Problem I.5.2 Application of Classical Integrators I.6 Exercises II.Numerical Integrators II.1 Runge-Kutta and Collocation Methods II.1.1 Runge-Kutta Methods II.1.2 Collocation Methods II.1.3 Gauss and Lobatto Collocation II.1.4 Discontinuous Collocation Methods II.2 Partitioned Runge-Kutta Methods II.2.1 Definition and First Examples II.2.2 Lobatto IIIA-IIIB Pairs II.2.3 Nystr6m Methods II.3 The Adjoint of a Method II.4 Composition Methods II.5 Splitting Methods II.6 Exercises III.Order Conditions, Trees and B-Series III.1 Runge-Kutta Order Conditions and B-Series III.1.1 Derivation of the Order Conditions III.1.2 B-Series III.1.3 Composition of Methods III.1.4 Composition of B-Series III.l.5 The Butcher Group III.2 Order Conditions for Partitioned Runge-Kutta Methods III.2.1 Bi-Coloured Trees and P-Series III.2.2 Order Conditions for Partitioned Runge-Kutta Methods III.2.3 Order Conditions for Nystrom Methods III.3 Order Conditions for Composition Methods III.3.1 Introduction III.3.2 The General Case III.3.3 Reduction of the Order Conditions III.3.4 Order Conditions for Splitting Methods III.4 The Baker-Campbell-Hausdorff Formula III.4.1 Derivative of the Exponential and Its Inverse III.4.2 The BCH Formula
II1.5 Order Conditions via the BCH Formula 1II.5.1 Calculus of Lie Derivatives III.5.2 Lie Brackets and Commutativity III.5.3 Splitting Methods II1.5.4 Composition Methods III.6 Exercises IV.Conservation of First Integrals and Methods on Manifolds IV.1 Examples of First Integrals IV.2 Quadratic Invariants IV.2.1 Runge-Kutta Methods IV.2.2 Partitioned Runge-Kutta Methods IV.2.3 Nystrom Methods IV.3 Polynomial Invariants IV.3.1 The Determinant as a First Integral IV.3.2 Isospectral Flows IV.4 Projection Methods IV.5 Numerical Methods Based on Local Coordinates IV.5.1 Manifolds and the Tangent Space IV.5.2 Differential Equations on Manifolds IV.5.3 Numerical Integrators on Manifolds IV.6 Differential Equations on Lie Groups 1V.7 Methods Based on the Magnus Series Expansion IV.8 Lie Group Methods IV.8.1 Crouch-Grossman Methods IV.8.2 Munthe-Kaas Methods IV.8.3 Further Coordinate Mappings IV.9 Geometric Numerical Integration Meets Geometric Numerical Linear Algebra IV.9.1 Numerical Integration on the Stiefel Manifold IV.9.2 Differential Equations on the Grassmann Manifold IV.9.3 Dynamical Low-Rank Approximation IV.10 Exercises V.Symmetric Integration and Reversibility VI.Symplectic Integration of Hamiltonian Systems VII.Non-Canonical Hamiitonian Systems VIII.Structure-Preserving Implementation IX.Backward Error Analysis and Structure Preservation X.Hamiltonian Perturbation Theory and Symplectic Integrators XI.Reversible Perturbation Theory and Symmetric Integrators XII.Dissivativeiv Perturbed Hamiltonian and Reversible Systems XIII.Oscillatory Differential Equations with Constant High Frequencies XIV.Oscillatory Differential Equations with Varying High Frequencies XV.Dynamics of Multistep Methods Bibliography Index
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