Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations (Springer Series in Computational Mathematics Book 31) 2nd Edition by Ernst Hairer (PDF)

Description

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 atheory 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.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐Excellent introduction to this topic, especially for folks with basic ODE knowledge. Also good for those wanting to potentially apply these methods to ML problems

⭐Very nice introduction to geometric numerical methods/symplectic methods

⭐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!

⭐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.

⭐I did not read the entire book, but the first few chapters are a good introduction to numerical integration methods.

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