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Table of Contents
Intro; Preface; Contents; Contributors; Why Geometric Numerical Integration?; 1 The Purpose of GNI; 2 The Story So Far; 2.1 Symplectic Integration; 2.2 Lie-Group Methods; 2.3 Conservation of Volume; 2.4 Preserving Energy and Other First Integrals; 3 Four Recent Stories of GNI; 3.1 Highly Oscillatory Hamiltonian Systems; 3.2 Kahan's `Unconventional' Method; 3.3 Applications to Celestial Mechanics; 3.4 Symmetric Zassenhaus Splitting and the Equations of Quantum Mechanics; 4 Beyond GNI; 4.1 GNI Meets Abstract Algebra; 4.2 Highly Oscillatory Quadrature; 4.3 Structured Linear Algebra; References
Lie Group Integrators1 Introduction; 2 The Setup; 3 Types of Schemes; 3.1 Schemes of Munthe-Kaas Type; 3.2 Integrators Based on Compositions of Flows; 4 Choice of Lie Group Actions; 4.1 Lie Group Acting on Itself by Multiplication; 4.2 The Affine Group and Its Use in Semilinear PDE Methods; 4.3 The Coadjoint Action and Lie-Poisson Systems; 4.4 Homogeneous Spaces and the Stiefel and Grassmann Manifolds; 4.5 Isospectral Flows; 4.6 Tangent and Cotangent Bundles; 5 Isotropy; 6 Order Theory for Lie Group Integrators; 6.1 Order Conditions for Commutator-Free Lie Group Integrators
6.2 Selecting a Minimal Set of Conditions7 Symplectic Lie Group Integrators; 7.1 Variational Integrators on Lie Groups; 8 Preservation of First Integrals; References; Lie-Butcher Series, Geometry, Algebra and Computation; 1 Introduction; 2 Geometry of Lie-Butcher Series; 2.1 Parallel Transport; 2.2 The Flat Cartan Connection on a Lie Group; 2.3 Numerical Integration; 3 Algebraic Structures of Lie-Butcher Theory; 3.1 Algebras; 3.2 Morphisms and Free Objects; 3.3 Enveloping Algebras; 4 Computing with Lie-Butcher Series; 4.1 Operations on Infinite Series Computed by Dualisation
4.2 Operations on Forests Computed by Recursions in a Magma4.3 Combinatorial Functions on Ordered Forests; 4.4 Concatenation and De-concatenation; 4.5 Shuffle and De-shuffle; 4.6 Grafting, Pruning, GL Product and GL Coproduct; 4.7 Substitution, Co-substitution, Scaling and Derivation; 4.8 Exponentials and Logarithms; 5 Concluding Remarks; 5.1 Programming in Haskell; References; Averaging and Computing Normal Forms with Word Series Algorithms; 1 Introduction; 2 Word Series; 2.1 Defining Word Series; 2.2 The Convolution Product; 2.3 Universal Formulations
3 Averaging of Quasiperiodically Forced Systems3.1 The Solution of the Oscillatory Problem; 3.2 The Transport Equation; 3.3 The Averaged System and the Change of Variables; 3.4 Geometric Properties; 3.5 Finding the Coefficients; 3.6 Changing the Initial Time; 4 Autonomous Problems; 4.1 Perturbed Problems; 4.2 The Transport Equation. Normal Forms; 5 Further Extensions; 5.1 Extended Word Series; 5.2 More General Perturbed Problems; References; Combinatorial Hopf Algebras for Interconnected Nonlinear Input-Output Systems with a View Towards Discretization; 1 Introduction; 2 Preliminaries
Lie Group Integrators1 Introduction; 2 The Setup; 3 Types of Schemes; 3.1 Schemes of Munthe-Kaas Type; 3.2 Integrators Based on Compositions of Flows; 4 Choice of Lie Group Actions; 4.1 Lie Group Acting on Itself by Multiplication; 4.2 The Affine Group and Its Use in Semilinear PDE Methods; 4.3 The Coadjoint Action and Lie-Poisson Systems; 4.4 Homogeneous Spaces and the Stiefel and Grassmann Manifolds; 4.5 Isospectral Flows; 4.6 Tangent and Cotangent Bundles; 5 Isotropy; 6 Order Theory for Lie Group Integrators; 6.1 Order Conditions for Commutator-Free Lie Group Integrators
6.2 Selecting a Minimal Set of Conditions7 Symplectic Lie Group Integrators; 7.1 Variational Integrators on Lie Groups; 8 Preservation of First Integrals; References; Lie-Butcher Series, Geometry, Algebra and Computation; 1 Introduction; 2 Geometry of Lie-Butcher Series; 2.1 Parallel Transport; 2.2 The Flat Cartan Connection on a Lie Group; 2.3 Numerical Integration; 3 Algebraic Structures of Lie-Butcher Theory; 3.1 Algebras; 3.2 Morphisms and Free Objects; 3.3 Enveloping Algebras; 4 Computing with Lie-Butcher Series; 4.1 Operations on Infinite Series Computed by Dualisation
4.2 Operations on Forests Computed by Recursions in a Magma4.3 Combinatorial Functions on Ordered Forests; 4.4 Concatenation and De-concatenation; 4.5 Shuffle and De-shuffle; 4.6 Grafting, Pruning, GL Product and GL Coproduct; 4.7 Substitution, Co-substitution, Scaling and Derivation; 4.8 Exponentials and Logarithms; 5 Concluding Remarks; 5.1 Programming in Haskell; References; Averaging and Computing Normal Forms with Word Series Algorithms; 1 Introduction; 2 Word Series; 2.1 Defining Word Series; 2.2 The Convolution Product; 2.3 Universal Formulations
3 Averaging of Quasiperiodically Forced Systems3.1 The Solution of the Oscillatory Problem; 3.2 The Transport Equation; 3.3 The Averaged System and the Change of Variables; 3.4 Geometric Properties; 3.5 Finding the Coefficients; 3.6 Changing the Initial Time; 4 Autonomous Problems; 4.1 Perturbed Problems; 4.2 The Transport Equation. Normal Forms; 5 Further Extensions; 5.1 Extended Word Series; 5.2 More General Perturbed Problems; References; Combinatorial Hopf Algebras for Interconnected Nonlinear Input-Output Systems with a View Towards Discretization; 1 Introduction; 2 Preliminaries