B-Series: Algebraic Analysis of Numerical Methods (Springer Series in Computational Mathematics, 55)

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