This book covers numerical methods that preserve properties of Hamiltonian systems, reversible systems, differential equations on manifolds and problems with highly oscillatory solutions. It presents a theory of symplectic and symmetric methods, which include various specially designed integrators, as well as discusses their construction and practical merits. The long-time behavior of the numerical solutions is studied using a backward error analysis combined with KAM theory.
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Editorial Reviews
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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)
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5 of 5 people found the following review helpful:
5.0 out of 5 stars
The Numerical Bible,
By Cybertronian (Finland) - See all my reviews
This review is from: Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations (Springer Series in Computational Mathematics) (Hardcover)
Why be interested in geometric numerical integrators you ask? Well, if you want algorithms that are accurate and conserve geometric properties of the dynamical flow you have to look at geometric integrators. For instance, in Hamiltonian systems the flow preserves the symplectic two-form (the volume of the phase space), and symplectic integrators do exactly that, which means that these algorithms are more accurate than their non-symplectic counterparts: conserved quantities are really conserved numerically.
Geometric Numerical Integration deals with the foundations, examples and actual applications of geometric integrators in various fields of research, and there is a lot on the more abstract theory of numerical mathematics, the classification of algorithms, provided with lots of mathematical and physical background needed to understand what is special about certain algorithms and advice on when, where and how to use them. It is completely self-contained, up-to-date, clear, well written, it has many references, and it is aimed at students and scientist who want to learn more about everything there is to know on geometric integrators. Admittedly, it is not completely inexpensive, but considering it is probably the only book you'll ever have to buy on geometric numerical integration and the fact that it looks great and is made very well, it is well worth the money!
1 of 1 people found the following review helpful:
5.0 out of 5 stars
The Bible it is...,
This review is from: Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations (Springer Series in Computational Mathematics) (Paperback)
To put it short: Anything you ever wanted to know about numerical integration of ordinary differential equations.
Accurate, complete and focused on the underlying ideas it is the perfect guide through the jungle of numerical methods for solving ODEs.
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Inside This Book
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Key Phrases - Statistically Improbable Phrases (SIPs):
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integrable reversible system, mollified impulse method, tangent space parametrization, discontinuous collocation method, conjugate symplecticity, count equal trees, parasitic solution components, consistent linear multistep method, modified differential equation, symmetric multistep methods, linear error growth, stable multistep methods, step size implementation, perturbed integrable systems, symplecticity condition, general linear methods, multiple time stepping, reversible analogue, formal first integral, symmetric composition methods, adiabatic variables, quadratic first integrals, implicit midpoint rule, step size function, symplectic methods Key Phrases - Capitalized Phrases (CAPs): (learn more)
Integrability Lemma, Feng Kang, Switching Lemma, Darboux-Lie Theorem, Triple Jump, Almost-Invariants of the Modulated Fourier Expansions, Numerical Methods Based, Solving Oscillatory Equations of Motion, Structure-Preserving Step Size Control, Towards Longer Time Steps, Newton's Principia New!
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integrable reversible system, mollified impulse method, tangent space parametrization, discontinuous collocation method, conjugate symplecticity, count equal trees, parasitic solution components, consistent linear multistep method, modified differential equation, symmetric multistep methods, linear error growth, stable multistep methods, step size implementation, perturbed integrable systems, symplecticity condition, general linear methods, multiple time stepping, reversible analogue, formal first integral, symmetric composition methods, adiabatic variables, quadratic first integrals, implicit midpoint rule, step size function, symplectic methods Key Phrases - Capitalized Phrases (CAPs): (learn more)
Integrability Lemma, Feng Kang, Switching Lemma, Darboux-Lie Theorem, Triple Jump, Almost-Invariants of the Modulated Fourier Expansions, Numerical Methods Based, Solving Oscillatory Equations of Motion, Structure-Preserving Step Size Control, Towards Longer Time Steps, Newton's Principia New!
Books on Related Topics | Concordance | Text Stats Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
Citations (learn more)
This book cites 24
books:
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books
cite this book:
- Lectures on Celestial Mechanics (Classics in Mathematics) by Carl L. Siegel on page 180, and Back Matter
- Linear Algebra (Undergraduate Texts in Mathematics) by Larry Smith on page 131, and Back Matter
- From Newton to Chaos: Modern Techniques for Understanding and Coping with Chaos in N-Body Dynamical Systems (NATO Science Series B: Physics) by Archie E. Roy in Back Matter
- Perturbation Theory for Linear Operators (Classics in Mathematics) by Tosio Kato in Back Matter
- Reversible Systems (Lecture Notes in Mathematics) by Mikhail B. Sevryuk on page 437
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