Ⅰ．Examples and Numerical Experiments

Ⅰ.1 First Problems and Methods

Ⅰ.1.1 The Lotka-Volterra Model

Ⅰ.1.2 First Numerical Methods

Ⅰ.1.3 The Pendulum as a Hamiltonian System

Ⅰ.1.4 The St？rmer-Verlet Scheme

Ⅰ.2 The Kepler Problem and the Outer Solar System

Ⅰ.2.1 Angular Momentum and Kepler's Second Law

Ⅰ.2.2 Exact Integration of the Kepler Problem

Ⅰ.2.3 Numerical Integration of the Kepler Problem

Ⅰ.2.4 The Outer Solar System

Ⅰ.3 The Hénon-Heiles Model

Ⅰ.4 Molecular Dynamics

Ⅰ.5 Highly Oscillatory Problems

Ⅰ.5.1 A Fermi-Pasta-Ulam Problem

Ⅰ.5.2 Application of Classical Integrators

Ⅰ.6 Exercises

Ⅱ．Numerical Integrators

Ⅱ.1 Runge-Kutta and Collocation Methods

Ⅱ.1.1 Runge-Kutta Methods

Ⅱ.1.2 Collocation Methods

Ⅱ.1.3 Gauss and Lobatto Collocation

Ⅱ.1.4 Discontinuous Collocation Methods

Ⅱ.2 Partitioned Runge-Kutta Methods

Ⅱ.2.1 Definition and First Examples

Ⅱ.2.2 Lobatto ⅢA-ⅢB Pairs

Ⅱ.2.3 Nystr？m Methods

Ⅱ.3 The Adjoint of a Method

Ⅱ.4 Composition Methods

Ⅱ.5 Splitting Methods

Ⅱ.6 Exercises

Ⅲ．Order Conditions,Trees and B-Series

Ⅲ.1 Runge-Kutta Order Conditions and B-Series

Ⅲ.1.1 Derivation of the Order Conditions

Ⅲ.1.2 B-Series

Ⅲ.1.3 Composition of Methods

Ⅲ.1.4 Composition of B-Series

Ⅲ.1.5 The Butcher Group

Ⅲ.2 Order Conditions for Partitioned Runge-Kutta Methods

Ⅲ.2.1 Bi-Coloured Trees and P-Series

Ⅲ.2.2 Order Conditions for Partitioned Runge-Kutta Methods

Ⅲ.2.3 OrderConditions for Nystr？m Methods

Ⅲ.3 Order Conditions for Composition Methods

Ⅲ.3.1 Introduction

Ⅲ.3.2 The General Case

Ⅲ.3.3 Reduction of the Order Conditions

Ⅲ.3.4 Order Conditions for Splitting Methods

Ⅲ.4 The Baker-Campbell-Hausdorff Formula

Ⅲ.4.1 Derivative of the Exponential and Its Inverse

Ⅲ.4.2 The BCH Formula

Ⅲ.5 Order Conditions via the BCH Formula

Ⅲ.5.1 Calculus of Lie Derivatives

Ⅲ.5.2 Lie Brackets and Commutativity

Ⅲ.5.3 Splitting Methods

Ⅲ.5.4 Composition Methods

Ⅲ.6 Exercises

Ⅳ．Conservation of First Integrals and Methods on Manifolds

Ⅳ.1 Examples of First Integrals

Ⅳ.2 Quadratic Invariants

Ⅳ.2.1 Runge-Kutta Methods

Ⅳ.2.2 Partitioned Runge-Kutta Methods

Ⅳ.2.3 Nystr？m Methods

Ⅳ.3 Polynomial Invariants

Ⅳ.3.1 The Determinant as a First Integral

Ⅳ.3.2 Isospectral Flows

Ⅳ.4 Projection Methods

Ⅳ.5 Numerical Methods Based on Local Coordinates

Ⅳ.5.1 Manifolds and the Tangent Space

Ⅳ.5.2 Differential Equations on Manifolds

Ⅳ.5.3 Numerical Integrators on Manifolds

Ⅳ.6 Differential Equations on Lie Groups

Ⅳ.7 Methods Based on the Magnus Series Expansion

Ⅳ.8 Lie Group Methods

Ⅳ.8.1 Crouch-Grossman Methods

Ⅳ.8.2 Munthe-Kaas Methods

Ⅳ.8.3 Further Coordinate Mappings

Ⅳ.9 Geometric Numerical Integration Meets Geometric Numerical Linear Algebra

Ⅳ.9.1 Numerical Integration on the Stiefel Manifold

Ⅳ.9.2 Differential Equations on the Grassmann Manifold

Ⅳ.9.3 Dynamical Low-Rank Approximation

Ⅳ.10 Exercises

Ⅴ．Symmetric Integration and Reversibility

Ⅴ.1 Reversible Differential Equations and Maps

Ⅴ.2 Symmetric Runge-Kutta Methods

Ⅴ.2.1 Collocation and Runge-Kutta Methods

Ⅴ.2.2 Partitioned Runge-Kutta Methods

Ⅴ.3 Symmetric Composition Methods

Ⅴ.3.1 Symmetric Composition of First Order Methods

Ⅴ.3.2 Symmetric Composition of Symmetric Methods

Ⅴ.3.3 Effective Order and Processing Methods

Ⅴ.4 Symmetric Methods on Manifolds

Ⅴ.4.1 Symmetric Projection

Ⅴ.4.2 Symmetric Methods Based on Local Coordinates

Ⅴ.5 Energy-Momentum Methods and Discrete Gradients

Ⅴ.6 Exercises

Ⅵ．Symplectic Integration of Hamiltonian Systems

Ⅵ.1 Hamiltonian Systems

Ⅵ.1.1 Lagrange's Equations

Ⅵ.1.2 Hamilton's Canonical Equations

Ⅵ.2 Symplectic Transformations

Ⅵ.3 First Examples of Symplectic Integrators

Ⅵ.4 Symplectic Runge-Kutta Methods

Ⅵ.4.1 Criterion of Symplecticity

Ⅵ.4.2 Connection Between Symplectic and Symmetric Methods

Ⅵ.5 Generating Functions

Ⅵ.5.1 Existence of Generating Functions

Ⅵ.5.2 Generating Function for Symplectic Runge-Kutta Methods

Ⅵ.5.3 The Hamilton-Jacobi Partial Differential Equation

Ⅵ.5.4 Methods Based on Generating Functions

Ⅵ.6 Variational Integrators

Ⅵ.6.1 Hamilton's Principle

Ⅵ.6.2 Discretization of Hamilton's Principle

Ⅵ.6.3 Symplectic Partitioned Runge-Kutta Methods Revisited

Ⅵ.6.4 Noether's Theorem

Ⅵ.7 Characterization of Symplectic Methods

Ⅵ.7.1 B-Series Methods Conserving Quadratic First Integrals

Ⅵ.7.2 Characterization of Symplectic P-Series（and B-Series）

Ⅵ.7.3 Irreducible Runge-Kutta Methods

Ⅵ.7.4 Characterization of Irreducible Symplectic Methods

Ⅵ.8 Conjugate Symplecticity

Ⅵ.8.1 Examples and Order Conditions

Ⅵ.8.2 Near Conservation of Quadratic First Integrals

Ⅵ.9 Volume Preservation

Ⅵ.10 Exercises

Ⅶ．Non-Canonical Hamiltonian Systems

Ⅶ.1 Constrained Mechanical Systems

Ⅶ.1.1 Introduction and Examples

Ⅶ.1.2 Hamiltonian Formulation

Ⅶ.1.3 A Symplectic First Order Method

Ⅶ.1.4 SHAKE and RATTLE

Ⅶ.1.5 The Lobatto ⅢA-ⅢB Pair

Ⅶ.1.6 Splitting Methods

Ⅶ.2 Poisson Systems

Ⅶ.2.1 Canonical Poisson Structure

Ⅶ.2.2 General Poisson Structures

Ⅶ.2.3 Hamiltonian Systems on Symplectic Submanifolds

Ⅶ.3 The Darboux-Lie Theorem

Ⅶ.3.1 Commutativity of Poisson Flows and Lie Brackets

Ⅶ.3.2 Simultaneous Linear Partial Differential Equations

Ⅶ.3.3 Coordinate Changes and the Darboux-Lie Theorem

Ⅶ.4 Poisson Integrators

Ⅶ.4.1 Poisson Maps and Symplectic Maps

Ⅶ.4.2 Poisson Integrators

Ⅶ.4.3 Integrators Based on the Darboux-Lie Theorem

Ⅶ.5 Rigid Body Dynamics and Lie-Poisson Systems

Ⅶ.5.1 History of the Euler Equations

Ⅶ.5.2 Hamiltonian Formulation of Rigid Body Motion

Ⅶ.5.3 Rigid Body Integrators

Ⅶ.5.4 Lie-poisson Systems

Ⅶ.5.5 Lie-Poisson Reduction

Ⅶ.6 Reduced Models of Quantum Dynamics

Ⅶ.6.1 Hamiltonian Structure of the Schr？dinger Equation

Ⅶ.6.2 The Dirac-Frenkel Variational Principle

Ⅶ.6.3 Gaussian Wavepacket Dynamics

Ⅶ.6.4 A Splitting Integrator for Gaussian Wavepackets

Ⅶ.7 Exercises

Ⅷ．Structure-Preserving Implementation

Ⅷ.1 Dangers of Using Standard Step Size Control

Ⅷ.2 Time Transformations

Ⅷ.2.1 Symplectic Integration

Ⅷ.2.2 Reversible Integration

Ⅷ.3 Structure-Preserving Step Size Control

Ⅷ.3.1 Proportional,Reversible Controllers

Ⅷ.3.2 Integrating,Reversible Controllers

Ⅷ.4 Multiple Time Stepping

Ⅷ.4.1 Fast-Slow Splitting：the Impulse Method

Ⅷ.4.2 Averaged Forces

Ⅷ.5 Reducing Rounding Errors

Ⅷ.6 Implementation of Implicit Methods

Ⅷ.6.1 Starting Approximations

Ⅷ.6.2 Fixed-Point Versus Newton Iteration

Ⅷ.7 Exercises

Ⅸ．Backward Error Analysis and Structure Preservation

Ⅸ.1 Modified Differential Equation-Examples

Ⅸ.2 Modified Equations of Symmetric Methods

Ⅸ.3 Modified Equations of Symplectic Methods

Ⅸ.3.1 Existence of a Local Modified Hamiltonian

Ⅸ.3.2 Existence of a Global Modified Hamiltonian

Ⅸ.3.3 Poisson Integrators

Ⅸ.4 Modified Equations of Splitting Methods

Ⅸ.5 Modified Equations of Methods on Manifolds

Ⅸ.5.1 Methods on Manifolds and First Integrals

Ⅸ.5.2 Constrained Hamiltonian Systems

Ⅸ.5.3 Lie-Poisson Integrators

Ⅸ.6 Modified Equations for Variable Step Sizes

Ⅸ.7 Rigorous Estimates-Local Error

Ⅸ.7.1 Estimation of the Derivatives of the Numerical Solution

Ⅸ.7.2 Estimation of the Coefficients of the Modified Equation

Ⅸ.7.3 Choice of N and the Estimation of the Local Error

Ⅸ.8 Long-Time Energy Conservation

Ⅸ.9 Modified Equation in Terms of Trees

Ⅸ.9.1 B-Series ofthe Modified Equation

Ⅸ.9.2 Elementary Hamiltonians

Ⅸ.9.3 Modified Hamiltonian

Ⅸ.9.4 First Integrals Close to the Hamiltonian

Ⅸ.9.5 Energy Conservation：Examples and Counter-Examples

Ⅸ.10 Extension to Partitioned Systems

Ⅸ.10.1 P-Series of the Modified Equation

Ⅸ.10.2 Elementary Hamiltonians

Ⅸ.11 Exercises

Ⅹ．Hamiltonian Perturbation Theory and Symplectic Integrators

Ⅹ.1 Completely Integrable Hamiltonian Systems

Ⅹ.1.1 Local Integration by Quadrature

Ⅹ.1.2 Completely Integrable Systems

Ⅹ.1.3 Action-Angle Variables

Ⅹ.1.4 Conditionally Periodic Flows

Ⅹ.1.5 The Toda Lattice-an Integrable System

Ⅹ.2 Transformations in the Perturbation Theory for Integrable Systems

Ⅹ.2.1 The Basic Scheme of Classical Perturbation Theory

Ⅹ.2.2 Lindstedt-Poincaré Series

Ⅹ.2.3 Kolmogorov's Iteration

Ⅹ.2.4 Birkhoff Normalization Near an Invariant Torus

Ⅹ.3 Linear Error Growth and Near-Preservation of First Integrals

Ⅹ.4 Near-Invariant Tori on Exponentially Long Times

Ⅹ.4.1 Estimates of Perturbation Series

Ⅹ.4.2 Near-Invariant Tori of Perturbed Integrable Systems

Ⅹ.4.3 Near-Invariant Tori of Symplectic Integrators

Ⅹ.5 Kolmogorov's Theorem on Invariant Tori

Ⅹ.5.1 Kolmogorov's Theorem

Ⅹ.5.2 KAM Tori under Symplectic Discretization

Ⅹ.6 Invariant Tori of Symplectic Maps

Ⅹ.6.1 A KAM Theorem for Symplectic Near-Identity Maps

Ⅹ.6.2 Invariant Tori of Symplectic Integrators

Ⅹ.6.3 Strongly Non-Resonant Step Sizes

Ⅹ.7 Exercises

Ⅺ．Reversible Perturbation Theory and Symmetric Integrators

Ⅺ.1 Integrable Reversible Systems

Ⅺ.2 Transformations in Reversible Perturbation Theory

Ⅺ.2.1 The Basic Scheme of Reversible Perturbation Theory

Ⅺ.2.2 Reversible Perturbation Series

Ⅺ.2.3 Reversible KAM Theory

Ⅺ.2.4 Reversible Birkhoff-Type Normalization

Ⅺ.3 Linear Error Growth and Near-Preservation of First Integrals

Ⅺ.4 Invariant Tori underReversible Discretization

Ⅺ.4.1 Near-Invariant Tori over Exponentially Long Times

Ⅺ.4.2 A KAM Theorem for Reversible Near-Identity Maps

Ⅺ.5 Exercises

Ⅻ．Dissipatively Perturbed Hamiltonian and Reversible Systems

Ⅻ.1 Numerical Experiments with Van der Pol's Equation

Ⅻ.2 Averaging Transformations

Ⅻ.2.1 The Basic Scheme of Averaging

Ⅻ.2.2 Perturbation Series

Ⅻ.3 Attractive Invariant Manifolds

Ⅻ.4 Weakly Attractive Invariant Tori of Perturbed Integrable Systems

Ⅻ.5 Weakly Attractive Invariant Tori of Numerical Integrators

Ⅻ.5.1 Modified Equations of Perturbed Differential Equations

Ⅻ.5.2 Symplectic Methods

Ⅻ.5.3 Symmetric Methods

Ⅻ.6 Exercises

ⅩⅢ．Oscillatory Differential Equations with Constant High Frequencies

ⅩⅢ.1 Towards Longer Time Steps in Solving Oscillatory Equations of Motion

ⅩⅢ.1.1 The St？rmer-Verlet Method vs.Multiple Time Scales

ⅩⅢ.1.2 Gautschi's and Deuflhard's Trigonometric Methods

ⅩⅢ.1.3 The Impulse Method

ⅪⅢ.1.4 The Mollified Impulse Method

ⅩⅢ.1.5 Gautschi's Method Revisited

ⅩⅢ.1.6 Two-Force Methods

ⅩⅢ.2 A Nonlinear Model Problem and Numerical Phenomena

ⅩⅢ.2.1 Time Scales in the Fermi-Pasta-Ulam Problem

ⅩⅢ.2.2 Numerical Methods

ⅩⅢ.2.3 Accuracy Comparisons

ⅩⅢ.2.4 Energy Exchange between Stiff Components

ⅩⅢ.2.5 Near-Conservation of Total and Oscillatory Energy

ⅩⅢ.3 Principal Terms of the Modulated Fourier Expansion

ⅩⅢ.3.1 Decomposition of the Exact Solution

ⅩⅢ.3.2 Decomposition of the Numerical Solution

ⅩⅢ.4 Accuracy and Slow Exchange

ⅩⅢ.4.1 Convergence Properties on Bounded Time Intervals

ⅩⅢ.4.2 Intra-Oscillatory and Oscillatory-Smooth Exchanges

ⅩⅢ.5 Modulated Fourier Expansions

ⅩⅢ.5.1 Expansion of the Exact Solution

ⅩⅢ.5.2 Expansion of the Numerical Solution

ⅩⅢ.5.3 Expansion of the Velocity Approximation

ⅩⅢ.6 Almost-Invariants of the Modulated Fourier Expansions

ⅪⅢ.6.1 The Hamiltonian of the Modulated Fourier Expansion

ⅩⅢ.6.2 A Formal Invariant Close to the Oscillatory Energy

ⅩⅢ.6.3 Almost-Invariants of the Numerical Method

ⅩⅢ.7 Long-Time Near-Conservation of Total and Oscillatory Energy

ⅩⅢ.8 Energy Behaviour of the St？rmer-Verlet Method

ⅩⅢ.9 Systems with Several Constant Frequencies

ⅩⅢ.9.1 Oscillatory Energies and Resonances

ⅩⅢ.9.2 Multi-Frequency Modulated Fourier Expansions

ⅩⅢ.9.3 Almost-Invariants of the Modulation System

ⅩⅢ.9.4 Long-Time Near-Conservation of Total and Oscillatory Energies

ⅩⅢ.10 Systems with Non-Constant Mass Matrix

ⅩⅢ.11 Exercises

Ⅹ．Oscillatory Differential Equations with Varying High Frequencies

ⅩⅣ.1 Linear Systems with Time-Dependent Skew-Hermitian Matrix

ⅩⅣ.1.1 Adiabatic Transformation and Adiabatic Invariants

ⅩⅣ.1.2 Adiabatic Integrators

ⅩⅣ.2 Mechanical Systems with Time-Dependent Frequencies

ⅩⅣ.2.1 Canonical Transformation to Adiabatic Variables

ⅩⅣ.2.2 Adiabatic Integrators

ⅩⅣ.2.3 Error Analysis of the Impulse Method

ⅩⅣ.2.4 Error Analysis of the Mollified Impulse Method

ⅩⅣ.3 Mechanical Systems with Solution-Dependent Frequencies

ⅩⅣ.3.1 Constraining Potentials

ⅩⅣ.3.2 Transformation to Adiabatic Variables

ⅩⅣ.3.3 Integrators in Adiabatic Variables

ⅩⅣ.3.4 Analysis of Multiple Time-Stepping Methods

ⅩⅣ.4 Exercises

ⅩⅤ．Dynamics of Multistep Methods

ⅩⅤ.1 Numerical Methods and Experiments

ⅩⅤ.1.1 Linear Multistep Methods

ⅩⅤ.1.2 Multistep Methods for Second Order Equations

ⅩⅤ.1.3 Partitioned Multistep Methods

ⅩⅤ.2 The Underlying One-Step Method

ⅩⅤ.2.1 Strictly Stable Multistep methods

ⅩⅤ.2.2 Formal Analysis for Weakly Stable Methods

ⅩⅤ.3 Backward Error Analysis

ⅩⅤ.3.1 Modified Equation for Smooth Numerical Solutions

ⅩⅤ.3.2 Parasitic Modified Equations

ⅩⅤ.4 Can Multistep Methods be Symplectic？

ⅩⅤ.4.1 Non-Symplecticity of the Underlying One-Step Method

ⅩⅤ.4.2 Symplecticity in the Higher-Dimensional Phase Space

ⅩⅤ.4.3 Modified Hamiltonian of Multistep Methods

ⅩⅤ.4.4 Modified Quadratic First Integrals

ⅩⅤ.5 Long-Term Stability

ⅩⅤ.5.1 Role of Growth Parameters

ⅩⅤ.5.2 Hamiltonian of the Full Modified System

ⅩⅤ.5.3 Long-Time Bounds for Parasitic Solution Components

ⅩⅤ.6 Explanation of the Long-Time Behaviour

ⅩⅤ.6.1 Conservation of Energy and Angular Momentum

ⅩⅤ.6.2 Linear Error Growth for Integrable Systems

ⅩⅤ.7 Practical Considerations

ⅩⅤ.7.1 Numerical Instabilities and Resonances

ⅩⅤ.7.2 Extension to Variable Step Sizes

ⅩⅤ.8 Multi-Value or General Linear Methods

ⅩⅤ.8.1 Underlying One-Step Method and Backward Error Analysis

ⅩⅤ.8.2 Symplecticity and Symmetry

ⅩⅤ.8.3 Growth Parameters

ⅩⅤ.9 Exercises

Bibliography

Index