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Geometric Numerical Integration

Structure-Preserving Algorithms for Ordinary Differential Equations

Part of the book series: Springer Series in Computational Mathematics (SSCM, volume 31)

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  • ISBN: 978-3-540-30666-5
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Table of contents (15 chapters)

  1. Front Matter

    Pages I-XVII
  2. Examples and Numerical Experiments

    • Ernst Hairer, Gerhard Wanner, Christian Lubich
    Pages 1-26
  3. Numerical Integrators

    • Ernst Hairer, Gerhard Wanner, Christian Lubich
    Pages 27-50
  4. Order Conditions, Trees and B-Series

    • Ernst Hairer, Gerhard Wanner, Christian Lubich
    Pages 51-96
  5. Conservation of First Integrals and Methods on Manifolds

    • Ernst Hairer, Gerhard Wanner, Christian Lubich
    Pages 97-142
  6. Symmetric Integration and Reversibility

    • Ernst Hairer, Gerhard Wanner, Christian Lubich
    Pages 143-178
  7. Symplectic Integration of Hamiltonian Systems

    • Ernst Hairer, Gerhard Wanner, Christian Lubich
    Pages 179-236
  8. Non-Canonical Hamiltonian Systems

    • Ernst Hairer, Gerhard Wanner, Christian Lubich
    Pages 237-302
  9. Structure-Preserving Implementation

    • Ernst Hairer, Gerhard Wanner, Christian Lubich
    Pages 303-336
  10. Backward Error Analysis and Structure Preservation

    • Ernst Hairer, Gerhard Wanner, Christian Lubich
    Pages 337-388
  11. Hamiltonian Perturbation Theory and Symplectic Integrators

    • Ernst Hairer, Gerhard Wanner, Christian Lubich
    Pages 389-436
  12. Reversible Perturbation Theory and Symmetric Integrators

    • Ernst Hairer, Gerhard Wanner, Christian Lubich
    Pages 437-454
  13. Dissipatively Perturbed Hamiltonian and Reversible Systems

    • Ernst Hairer, Gerhard Wanner, Christian Lubich
    Pages 455-470
  14. Oscillatory Differential Equations with Constant High Frequencies

    • Ernst Hairer, Gerhard Wanner, Christian Lubich
    Pages 471-530
  15. Oscillatory Differential Equations with Varying High Frequencies

    • Ernst Hairer, Gerhard Wanner, Christian Lubich
    Pages 531-565
  16. Dynamics of Multistep Methods

    • Ernst Hairer, Gerhard Wanner, Christian Lubich
    Pages 567-616
  17. Back Matter

    Pages 617-644

About this book

Numerical 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. The second edition is substantially revised and enlarged, with many improvements in the presentation and additions concerning in particular non-canonical Hamiltonian systems, highly oscillatory mechanical systems, and the dynamics of multistep methods.

Keywords

  • Hamiltonian and reversible systems
  • Numerical integration
  • algorithms
  • calculus
  • differential equations on manifolds
  • geometric numerical integration
  • symplectic and symmetric methods

Reviews

From the reviews of the second edition:

"This book is highly recommended for advanced courses in numerical methods for ordinary differential equations as well as a reference for researchers/developers in the field of geometric integration, differential equations in general and related subjects. It is a must for academic and industrial libraries." -- MATHEMATICAL REVIEWS

"The second revised edition of the monograph is a fine work organized in fifteen chapters, updated and extended. … The material of the book is organized in sections which are … self-contained, so that one can dip into the book to learn a particular topic … . A person interested in geometrical numerical integration will find this book extremely useful." (Calin Ioan Gheorghiu, Zentralblatt MATH, Vol. 1094 (20), 2006)

Authors and Affiliations

  • Section de Mathématiques, Université de Genève, Genève 4, Switzerland

    Ernst Hairer, Gerhard Wanner

  • Mathematisches Institut, Universität Tübingen, Tübingen, Germany

    Christian Lubich

Bibliographic Information

Buying options

eBook USD 189.00
Price excludes VAT (USA)
  • ISBN: 978-3-540-30666-5
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book USD 249.99
Price excludes VAT (USA)
Hardcover Book USD 249.99
Price excludes VAT (USA)