Numerical Methods for Ordinary Differential Equations 2nd Edition by J. C. Butcher (PDF)

Description

In recent years the study of numerical methods for solving ordinary differential equations has seen many new developments. This second edition of the author’s pioneering text is fully revised and updated to acknowledge many of these developments. It includes a complete treatment of linear multistep methods whilst maintaining its unique and comprehensive emphasis on Runge-Kutta methods and general linear methods. Although the specialist topics are taken to an advanced level, the entry point to the volume as a whole is not especially demanding. Early chapters provide a wide-ranging introduction to differential equations and difference equations together with a survey of numerical differential equation methods, based on the fundamental Euler method with more sophisticated methods presented as generalizations of Euler.Features of the book include Introductory work on differential and difference equations. A comprehensive introduction to the theory and practice of solving ordinary differential equations numerically. A detailed analysis of Runge-Kutta methods and of linear multistep methods. A complete study of general linear methods from both theoretical and practical points of view. The latest results on practical general linear methods and their implementation. A balance between informal discussion and rigorous mathematical style. Examples and exercises integrated into each chapter enhancing the suitability of the book as a course text or a self-study treatise.Written in a lucid style by one of the worlds leading authorities on numerical methods for ordinary differential equations and drawing upon his vast experience, this new edition provides an accessible and self-contained introduction, ideal for researchers and students following courses on numerical methods, engineering and other sciences.

User’s Reviews

Editorial Reviews: From the Back Cover In recent years the study of numerical methods for solving ordinary differential equations has seen many new developments. This second edition of the author’s pioneering text is fully revised and updated to acknowledge many of these developments. It includes a complete treatment of linear multistep methods whilst maintaining its unique and comprehensive emphasis on Runge-Kutta methods and general linear methods. Although the specialist topics are taken to an advanced level, the entry point to the volume as a whole is not especially demanding. Early chapters provide a wide-ranging introduction to differential equations and difference equations together with a survey of numerical differential equation methods, based on the fundamental Euler method with more sophisticated methods presented as generalizations of Euler.Features of the book include Introductory work on differential and difference equations. A comprehensive introduction to the theory and practice of solving ordinary differential equations numerically. A detailed analysis of Runge-Kutta methods and of linear multistep methods. A complete study of general linear methods from both theoretical and practical points of view. The latest results on practical general linear methods and their implementation. A balance between informal discussion and rigorous mathematical style. Examples and exercises integrated into each chapter enhancing the suitability of the book as a course text or a self-study treatise.Written in a lucid style by one of the worlds leading authorities on numerical methods for ordinary differential equations and drawing upon his vast experience, this new edition provides an accessible and self-contained introduction, ideal for researchers and students following courses on numerical methods, engineering and other sciences.

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

⭐This book is outstanding. Anyone trying to get into the how’s and why’s of numerical ODE methods should definitely start here. Great review of Runge-Kutta methods, and I really like the introduction to linear multistep methods. Great stuff.

⭐If you have even a decent understanding of calculus and a typical first course in numerical analysis (which for most students is just a numerical cookbook of techniques and algorithms), this is absolutely the best “next step” for those who are getting into numerical computations for ODEs.If you have a strong understanding of numerical analysis as a whole, this book is still the best thing out there.Numerical analysis continues to grow, but with regards to the specific treatment of ODEs, methodologies don’t change too much. Sure, we develop new ideas that bring in the geometry of our ODEs into play or come up with more robust partitioning methods, or seek better path finding methods (all of which are nicely presented here, by the way!), but the foundations for “how” we even dream up such ideas is more or less the same.And that’s what this book is absolutely amazing with. Both in the clarity of presentation as well as the specific problems it chooses, this book forces you to think in terms of mathematical analysis instead of simple algorithmic analysis (where we prefer to find better bounds on efficiency instead of revisiting the fundamental idea).If you can work through this book–which I find ideal for self-study for the student who has either a strong foundation in numerical analysis or ordinary differential equations–you’re basically set for going through modern research into solving DEs in general and contributing your own without too much of a brick wall to stand in your way.

⭐This is the Second Edition of a classic work on solving differential equations numerically. The well-known author and teacher has published pathfinding papers on this topic from the early 1960’s to the present. The “Butcher Tableau” is the standard short-hand display for Runge-Kutta methods.The book sets the table by describing a set of standard problems for numerical solution. Then, as new methods are described their efficacy on the standard problems is graphed and tabulated.Explicit and implicit Runge-Kutta Methods, Linear Multistep Methods, Taylor Series Methods and General Linear Methods are thoroughly described from the viewpoint of accuracy and stability. Symplectic Runge-Kutta methods, ‘Almost’ Runge-Kutta methods and Inverse Runge-Kutta Methods are among the additional covered topics of current interest.Practical issues for the implementer of algorithms based on these methods, such as error estimation and step-size control, are well-covered. References to the topic’s literature are also outstanding.To use the book, a background in calculus and numerical analysis is essential.Current topics in this field are well-represented. The author has taught many of the present class of researchers, and the clarity of a great teacher is apparent.Web resources and problems at the end of each section enhance the book’s value for a semester-long course.

⭐An in-depth analysis from one of the best experts in the field of the latest developments in the numerical solution of ODEs with special emphasis on the Runge-Kutta methods, although Linear Multistep Methods and General Linear Methods are also thoroughly described. A must read for any researcher with interest on the numerical solution of initial value problems.

⭐Da un genio, nonché uno dei padri della teoria dei metodi numerici per ODEs, non poteva che venire uno dei migliori testi sull’argomento. L’esposizione è chiara fin dalle prime pagine e il lettore è portato nei meandri dei vari argomenti: metodi lineari multistep, metodi Runge-Kutta, inclusa la famigerata teoria dell’ordine ed infine è proposta la famiglia dei Metodi Generali Lineari, che costituiscono una pagina ampia, aperta ed affascinante del settore.

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