Description 
1 online resource (316 pages) 
Series 
Springer Series in Computational Mathematics ; v. 55 

Springer series in computational mathematics ; 55.

Contents 
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 RungeKutta methods  1.6 Multivalue methods  1.7 Bseries 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 Bseries and algebraic analysis  3.1 Introduction  3.2 Autonomous formulation and mappings  3.3 Fréchet derivatives and Taylor series  3.4 Elementary differentials and Bseries  3.5 Bseries for flow_h and implicit_h  3.6 Elementary weights and the order of RungeKutta methods  3.7 Elementary differentials based on Kronecker products  3.8 Attainable values of elementary weights and differentials  3.9 Composition of Bseries 

Chapter 4 Algebraic analysis and integration methods  4.1 Introduction  4.2 Integration methods  4.3 Equivalence and reducibility of RungeKutta methods  4.4 Equivalence and reducibility of integration methods  4.5 Compositions of RungeKutta methods  4.7 The Bgroup and subgroups  4.8 Linear operators on B* and B̂0  Chapter 5 Bseries and RungeKutta methods  5.1 Introduction  5.2 Order analysis for scalar problems  5.3 Stability of RungeKutta methods  5.4 Explicit RungeKutta methods  5.5 Attainable order of explicit methods  5.6 Implicit RungeKutta methods 

5.7 Effective order methods  Chapter 6 Bseries 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 Bseries and geometric integration  7.1 Introduction  7.2 Hamiltonian and related problems  7.3 Canonical and symplectic RungeKutta methods  7.4 Gsymplectic 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 
Summary 
Bseries, 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 Bseries for a clean and elegant approach to the derivation of high order and efficient methods. However, the utility of Bseries goes much further and opens a path to the design and construction of highly accurate and efficient multivalue methods. This book offers a selfcontained introduction to Bseries 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 Bseries 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 Bseries 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 Bseries approach is both natural and essential. The final chapter introduces some aspects of geometric integration, from a Bseries point of view. Placing Bseries 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 openended 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 selfstudy 
Bibliography 
Includes bibliographical references and index 
Notes 
Print version record 
Subject 
RungeKutta formulas.


Ecuaciones diferenciales


Dofferential equations


RungeKutta formulas

Form 
Electronic book

ISBN 
9783030709563 

3030709566 
