B-series : algebraic analysis of numerical methods / John C. Butcher.
Butcher, J. C. (John Charles), 1933-
2021
QA297.5
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Title B-series : algebraic analysis of numerical methods / John C. Butcher.
Author Butcher, J. C. (John Charles), 1933-
ISBN 9783030709563 (electronic bk.)
3030709566 (electronic bk.)
9783030709556 (print)
3030709558
DOI https://doi.org/10.1007/978-3-030-70956-3
Publication Details Cham : Springer, 2021.
Language English
Description 1 online resource (316 pages)
Item Number 10.1007/978-3-030-70956-3 doi
Call Number QA297.5
Dewey Decimal Classification 511/.4
Summary B-series, also known as Butcher series, are an algebraic tool for analysing solutions to ordinary differential equations, including approximate solutions. Through the formulation and manipulation of these series, properties of numerical methods can be assessed. Runge Kutta methods, in particular, depend on B-series for a clean and elegant approach to the derivation of high order and efficient methods. However, the utility of B-series goes much further and opens a path to the design and construction of highly accurate and efficient multivalue methods. This book offers a self-contained introduction to B-series by a pioneer of the subject. After a preliminary chapter providing background on differential equations and numerical methods, a broad exposition of graphs and trees is presented. This is essential preparation for the third chapter, in which the main ideas of B-series are introduced and developed. In chapter four, algebraic aspects are further analysed in the context of integration methods, a generalization of Runge Kutta methods to infinite index sets. Chapter five, on explicit and implicit Runge Kutta methods, contrasts the B-series and classical approaches. Chapter six, on multivalue methods, gives a traditional review of linear multistep methods and expands this to general linear methods, for which the B-series approach is both natural and essential. The final chapter introduces some aspects of geometric integration, from a B-series point of view. Placing B-series at the centre of its most important applications makes this book an invaluable resource for scientists, engineers and mathematicians who depend on computational modelling, not to mention computational scientists who carry out research on numerical methods in differential equations. In addition to exercises with solutions and study notes, a number of open-ended projects are suggested. This combination makes the book ideal as a textbook for specialised courses on numerical methods for differential equations, as well as suitable for self-study.
Bibliography, etc. Note Includes bibliographical references and index.
Access Note Access limited to authorized users.
Source of Description Description based on print version record.
Series Springer series in computational mathematics ; 55.
Available in Other Form B-Series.
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Intro
Foreword
Preface
Contents
Chapter 1 Differential equations, numerical methods and algebraic analysis
1.1 Introduction
1.2 Differential equations
1.3 Examples of differential equations
1.4 The Euler method
1.5 Runge-Kutta methods
1.6 Multivalue methods
1.7 B-series analysis of numerical methods
Chapter 2 Trees and forests
2.1 Introduction to trees, graphs and forests
2.2 Rooted trees and unrooted (free) trees
2.3 Forests and trees
2.4 Tree and forest spaces
2.5 Functions of trees
2.6 Trees, partitions and evolutions

2.7 Trees and stumps
2.8 Subtrees, supertrees and prunings
2.9 Antipodes of trees and forests
Chapter 3 B-series and algebraic analysis
3.1 Introduction
3.2 Autonomous formulation and mappings
3.3 Fréchet derivatives and Taylor series
3.4 Elementary differentials and B-series
3.5 B-series for flow_h and implicit_h
3.6 Elementary weights and the order of Runge-Kutta methods
3.7 Elementary differentials based on Kronecker products
3.8 Attainable values of elementary weights and differentials
3.9 Composition of B-series

Chapter 4 Algebraic analysis and integration methods
4.1 Introduction
4.2 Integration methods
4.3 Equivalence and reducibility of Runge-Kutta methods
4.4 Equivalence and reducibility of integration methods
4.5 Compositions of Runge-Kutta methods
4.7 The B-group and subgroups
4.8 Linear operators on B* and B^0
Chapter 5 B-series and Runge-Kutta methods
5.1 Introduction
5.2 Order analysis for scalar problems
5.3 Stability of Runge-Kutta methods
5.4 Explicit Runge-Kutta methods
5.5 Attainable order of explicit methods
5.6 Implicit Runge-Kutta methods

5.7 Effective order methods
Chapter 6 B-series and multivalue methods
6.1 Introduction
6.2 Survey of linear multistep methods
6.3 Motivations for general linear methods
6.4 Formulation of general linear methods
6.5 Order of general linear methods
6.6 An algorithm for determining order
Chapter 7 B-series and geometric integration
7.1 Introduction
7.2 Hamiltonian and related problems
7.3 Canonical and symplectic Runge-Kutta methods
7.4 G-symplectic methods
7.5 Derivation of a fourth order method
7.6 Construction of a sixth order method
7.7 Implementation

7.8 Numerical simulations
7.9 Energy preserving methods
Answers to the exercises
References
Index

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