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Ⅰ．Examples and Numerical Experiments1

Ⅰ.1 First Problems and Methods1

Ⅰ.1.1 The Lotka-Volterra Model1

Ⅰ.1.2 First Numerical Methods3

Ⅰ.1.3 The Pendulum as a Hamiltonian System4

Ⅰ.1.4 The St？rmer-Verlet Scheme7

Ⅰ.2 The Kepler Problem and the Outer Solar System8

Ⅰ.2.1 Angular Momentum and Kepler's Second Law9

Ⅰ.2.2 Exact Integration of the Kepler Problem10

Ⅰ.2.3 Numerical Integration of the Kepler Problem12

Ⅰ.2.4 The Outer Solar System13

Ⅰ.3 The Hénon-Heiles Model15

Ⅰ.4 Molecular Dynamics18

Ⅰ.5 Highly Oscillatory Problems21

Ⅰ.5.1 A Fermi-Pasta-Ulam Problem21

Ⅰ.5.2 Application of Classical Integrators23

Ⅰ.6 Exercises24

Ⅱ．Numerical Integrators27

Ⅱ.1 Runge-Kutta and Collocation Methods27

Ⅱ.1.1 Runge-Kutta Methods28

Ⅱ.1.2 Collocation Methods30

Ⅱ.1.3 Gauss and Lobatto Collocation34

Ⅱ.1.4 Discontinuous Collocation Methods35

Ⅱ.2 Partitioned Runge-Kutta Methods38

Ⅱ.2.1 Definition and First Examples38

Ⅱ.2.2 Lobatto ⅢA-ⅢB Pairs40

Ⅱ.2.3 Nystr？m Methods41

Ⅱ.3 The Adjoint of a Method42

Ⅱ.4 Composition Methods43

Ⅱ.5 Splitting Methods47

Ⅱ.6 Exercises50

Ⅲ．Order Conditions,Trees and B-Series51

Ⅲ.1 Runge-Kutta Order Conditions and B-Series51

Ⅲ.1.1 Derivation of the Order Conditions51

Ⅲ.1.2 B-Series56

Ⅲ.1.3 Composition of Methods59

Ⅲ.1.4 Composition of B-Series61

Ⅲ.1.5 The Butcher Group64

Ⅲ.2 Order Conditions for Partitioned Runge-Kutta Methods66

Ⅲ.2.1 Bi-Coloured Trees and P-Series66

Ⅲ.2.2 Order Conditions for Partitioned Runge-Kutta Methods68

Ⅲ.2.3 OrderConditions for Nystr？m Methods69

Ⅲ.3 Order Conditions for Composition Methods71

Ⅲ.3.1 Introduction71

Ⅲ.3.2 The General Case73

Ⅲ.3.3 Reduction of the Order Conditions75

Ⅲ.3.4 Order Conditions for Splitting Methods80

Ⅲ.4 The Baker-Campbell-Hausdorff Formula83

Ⅲ.4.1 Derivative of the Exponential and Its Inverse83

Ⅲ.4.2 The BCH Formula84

Ⅲ.5 Order Conditions via the BCH Formula87

Ⅲ.5.1 Calculus of Lie Derivatives87

Ⅲ.5.2 Lie Brackets and Commutativity89

Ⅲ.5.3 Splitting Methods91

Ⅲ.5.4 Composition Methods92

Ⅲ.6 Exercises95

Ⅳ．Conservation of First Integrals and Methods on Manifolds97

Ⅳ.1 Examples of First Integrals97

Ⅳ.2 Quadratic Invariants101

Ⅳ.2.1 Runge-Kutta Methods101

Ⅳ.2.2 Partitioned Runge-Kutta Methods102

Ⅳ.2.3 Nystr？m Methods104

Ⅳ.3 Polynomial Invariants105

Ⅳ.3.1 The Determinant as a First Integral105

Ⅳ.3.2 Isospectral Flows107

Ⅳ.4 Projection Methods109

Ⅳ.5 Numerical Methods Based on Local Coordinates113

Ⅳ.5.1 Manifolds and the Tangent Space114

Ⅳ.5.2 Differential Equations on Manifolds115

Ⅳ.5.3 Numerical Integrators on Manifolds116

Ⅳ.6 Differential Equations on Lie Groups118

Ⅳ.7 Methods Based on the Magnus Series Expansion121

Ⅳ.8 Lie Group Methods123

Ⅳ.8.1 Crouch-Grossman Methods124

Ⅳ.8.2 Munthe-Kaas Methods125

Ⅳ.8.3 Further Coordinate Mappings128

Ⅳ.9 Geometric Numerical Integration Meets Geometric Numerical Linear Algebra131

Ⅳ.9.1 Numerical Integration on the Stiefel Manifold131

Ⅳ.9.2 Differential Equations on the Grassmann Manifold135

Ⅳ.9.3 Dynamical Low-Rank Approximation137

Ⅳ.10 Exercises139

Ⅴ．Symmetric Integration and Reversibility143

Ⅴ.1 Reversible Differential Equations and Maps143

Ⅴ.2 Symmetric Runge-Kutta Methods146

Ⅴ.2.1 Collocation and Runge-Kutta Methods146

Ⅴ.2.2 Partitioned Runge-Kutta Methods148

Ⅴ.3 Symmetric Composition Methods149

Ⅴ.3.1 Symmetric Composition of First Order Methods150

Ⅴ.3.2 Symmetric Composition of Symmetric Methods154

Ⅴ.3.3 Effective Order and Processing Methods158

Ⅴ.4 Symmetric Methods on Manifolds161

Ⅴ.4.1 Symmetric Projection161

Ⅴ.4.2 Symmetric Methods Based on Local Coordinates166

Ⅴ.5 Energy-Momentum Methods and Discrete Gradients171

Ⅴ.6 Exercises176

Ⅵ．Symplectic Integration of Hamiltonian Systems179

Ⅵ.1 Hamiltonian Systems180

Ⅵ.1.1 Lagrange's Equations180

Ⅵ.1.2 Hamilton's Canonical Equations181

Ⅵ.2 Symplectic Transformations182

Ⅵ.3 First Examples of Symplectic Integrators187

Ⅵ.4 Symplectic Runge-Kutta Methods191

Ⅵ.4.1 Criterion of Symplecticity191

Ⅵ.4.2 Connection Between Symplectic and Symmetric Methods194

Ⅵ.5 Generating Functions195

Ⅵ.5.1 Existence of Generating Functions195

Ⅵ.5.2 Generating Function for Symplectic Runge-Kutta Methods198

Ⅵ.5.3 The Hamilton-Jacobi Partial Differential Equation200

Ⅵ.5.4 Methods Based on Generating Functions203

Ⅵ.6 Variational Integrators204

Ⅵ.6.1 Hamilton's Principle204

Ⅵ.6.2 Discretization of Hamilton's Principle206

Ⅵ.6.3 Symplectic Partitioned Runge-Kutta Methods Revisited208

Ⅵ.6.4 Noether's Theorem210

Ⅵ.7 Characterization of Symplectic Methods212

Ⅵ.7.1 B-Series Methods Conserving Quadratic First Integrals212

Ⅵ.7.2 Characterization of Symplectic P-Series（and B-Series）217

Ⅵ.7.3 Irreducible Runge-Kutta Methods220

Ⅵ.7.4 Characterization of Irreducible Symplectic Methods222

Ⅵ.8 Conjugate Symplecticity222

Ⅵ.8.1 Examples and Order Conditions223

Ⅵ.8.2 Near Conservation of Quadratic First Integrals225

Ⅵ.9 Volume Preservation227

Ⅵ.10 Exercises233

Ⅶ．Non-Canonical Hamiltonian Systems237

Ⅶ.1 Constrained Mechanical Systems237

Ⅶ.1.1 Introduction and Examples237

Ⅶ.1.2 Hamiltonian Formulation239

Ⅶ.1.3 A Symplectic First Order Method242

Ⅶ.1.4 SHAKE and RATTLE245

Ⅶ.1.5 The Lobatto ⅢA-ⅢB Pair247

Ⅶ.1.6 Splitting Methods252

Ⅶ.2 Poisson Systems254

Ⅶ.2.1 Canonical Poisson Structure254

Ⅶ.2.2 General Poisson Structures256

Ⅶ.2.3 Hamiltonian Systems on Symplectic Submanifolds258

Ⅶ.3 The Darboux-Lie Theorem261

Ⅶ.3.1 Commutativity of Poisson Flows and Lie Brackets261

Ⅶ.3.2 Simultaneous Linear Partial Differential Equations262

Ⅶ.3.3 Coordinate Changes and the Darboux-Lie Theorem265

Ⅶ.4 Poisson Integrators268

Ⅶ.4.1 Poisson Maps and Symplectic Maps268

Ⅶ.4.2 Poisson Integrators270

Ⅶ.4.3 Integrators Based on the Darboux-Lie Theorem272

Ⅶ.5 Rigid Body Dynamics and Lie-Poisson Systems274

Ⅶ.5.1 History of the Euler Equations275

Ⅶ.5.2 Hamiltonian Formulation of Rigid Body Motion278

Ⅶ.5.3 Rigid Body Integrators280

Ⅶ.5.4 Lie-poisson Systems286

Ⅶ.5.5 Lie-Poisson Reduction289

Ⅶ.6 Reduced Models of Quantum Dynamics293

Ⅶ.6.1 Hamiltonian Structure of the Schr？dinger Equation293

Ⅶ.6.2 The Dirac-Frenkel Variational Principle295

Ⅶ.6.3 Gaussian Wavepacket Dynamics296

Ⅶ.6.4 A Splitting Integrator for Gaussian Wavepackets298

Ⅶ.7 Exercises301

Ⅷ．Structure-Preserving Implementation303

Ⅷ.1 Dangers of Using Standard Step Size Control303

Ⅷ.2 Time Transformations306

Ⅷ.2.1 Symplectic Integration306

Ⅷ.2.2 Reversible Integration309

Ⅷ.3 Structure-Preserving Step Size Control310

Ⅷ.3.1 Proportional,Reversible Controllers310

Ⅷ.3.2 Integrating,Reversible Controllers314

Ⅷ.4 Multiple Time Stepping316

Ⅷ.4.1 Fast-Slow Splitting：the Impulse Method317

Ⅷ.4.2 Averaged Forces319

Ⅷ.5 Reducing Rounding Errors322

Ⅷ.6 Implementation of Implicit Methods325

Ⅷ.6.1 Starting Approximations326

Ⅷ.6.2 Fixed-Point Versus Newton Iteration330

Ⅷ.7 Exercises335

Ⅸ．Backward Error Analysis and Structure Preservation337

Ⅸ.1 Modified Differential Equation-Examples337

Ⅸ.2 Modified Equations of Symmetric Methods342

Ⅸ.3 Modified Equations of Symplectic Methods343

Ⅸ.3.1 Existence of a Local Modified Hamiltonian343

Ⅸ.3.2 Existence of a Global Modified Hamiltonian344

Ⅸ.3.3 Poisson Integrators347

Ⅸ.4 Modified Equations of Splitting Methods348

Ⅸ.5 Modified Equations of Methods on Manifolds350

Ⅸ.5.1 Methods on Manifolds and First Integrals350

Ⅸ.5.2 Constrained Hamiltonian Systems352

Ⅸ.5.3 Lie-Poisson Integrators354

Ⅸ.6 Modified Equations for Variable Step Sizes356

Ⅸ.7 Rigorous Estimates-Local Error358

Ⅸ.7.1 Estimation of the Derivatives of the Numerical Solution360

Ⅸ.7.2 Estimation of the Coefficients of the Modified Equation362

Ⅸ.7.3 Choice of N and the Estimation of the Local Error364

Ⅸ.8 Long-Time Energy Conservation366

Ⅸ.9 Modified Equation in Terms of Trees369

Ⅸ.9.1 B-Series ofthe Modified Equation369

Ⅸ.9.2 Elementary Hamiltonians373

Ⅸ.9.3 Modified Hamiltonian375

Ⅸ.9.4 First Integrals Close to the Hamiltonian375

Ⅸ.9.5 Energy Conservation：Examples and Counter-Examples379

Ⅸ.10 Extension to Partitioned Systems381

Ⅸ.10.1 P-Series of the Modified Equation381

Ⅸ.10.2 Elementary Hamiltonians384

Ⅸ.11 Exercises386

Ⅹ．Hamiltonian Perturbation Theory and Symplectic Integrators389

Ⅹ.1 Completely Integrable Hamiltonian Systems390

Ⅹ.1.1 Local Integration by Quadrature390

Ⅹ.1.2 Completely Integrable Systems393

Ⅹ.1.3 Action-Angle Variables397

Ⅹ.1.4 Conditionally Periodic Flows399

Ⅹ.1.5 The Toda Lattice-an Integrable System402

Ⅹ.2 Transformations in the Perturbation Theory for Integrable Systems404

Ⅹ.2.1 The Basic Scheme of Classical Perturbation Theory405

Ⅹ.2.2 Lindstedt-Poincaré Series406

Ⅹ.2.3 Kolmogorov's Iteration410

Ⅹ.2.4 Birkhoff Normalization Near an Invariant Torus412

Ⅹ.3 Linear Error Growth and Near-Preservation of First Integrals413

Ⅹ.4 Near-Invariant Tori on Exponentially Long Times417

Ⅹ.4.1 Estimates of Perturbation Series417

Ⅹ.4.2 Near-Invariant Tori of Perturbed Integrable Systems421

Ⅹ.4.3 Near-Invariant Tori of Symplectic Integrators422

Ⅹ.5 Kolmogorov's Theorem on Invariant Tori423

Ⅹ.5.1 Kolmogorov's Theorem423

Ⅹ.5.2 KAM Tori under Symplectic Discretization428

Ⅹ.6 Invariant Tori of Symplectic Maps430

Ⅹ.6.1 A KAM Theorem for Symplectic Near-Identity Maps431

Ⅹ.6.2 Invariant Tori of Symplectic Integrators433

Ⅹ.6.3 Strongly Non-Resonant Step Sizes433

Ⅹ.7 Exercises434

Ⅺ．Reversible Perturbation Theory and Symmetric Integrators437

Ⅺ.1 Integrable Reversible Systems437

Ⅺ.2 Transformations in Reversible Perturbation Theory442

Ⅺ.2.1 The Basic Scheme of Reversible Perturbation Theory443

Ⅺ.2.2 Reversible Perturbation Series444

Ⅺ.2.3 Reversible KAM Theory445

Ⅺ.2.4 Reversible Birkhoff-Type Normalization447

Ⅺ.3 Linear Error Growth and Near-Preservation of First Integrals448

Ⅺ.4 Invariant Tori underReversible Discretization451

Ⅺ.4.1 Near-Invariant Tori over Exponentially Long Times451

Ⅺ.4.2 A KAM Theorem for Reversible Near-Identity Maps451

Ⅺ.5 Exercises453

Ⅻ．Dissipatively Perturbed Hamiltonian and Reversible Systems455

Ⅻ.1 Numerical Experiments with Van der Pol's Equation455

Ⅻ.2 Averaging Transformations458

Ⅻ.2.1 The Basic Scheme of Averaging458

Ⅻ.2.2 Perturbation Series459

Ⅻ.3 Attractive Invariant Manifolds460

Ⅻ.4 Weakly Attractive Invariant Tori of Perturbed Integrable Systems464

Ⅻ.5 Weakly Attractive Invariant Tori of Numerical Integrators465

Ⅻ.5.1 Modified Equations of Perturbed Differential Equations466

Ⅻ.5.2 Symplectic Methods467

Ⅻ.5.3 Symmetric Methods469

Ⅻ.6 Exercises469

ⅩⅢ．Oscillatory Differential Equations with Constant High Frequencies471

ⅩⅢ.1 Towards Longer Time Steps in Solving Oscillatory Equations of Motion471

ⅩⅢ.1.1 The St？rmer-Verlet Method vs.Multiple Time Scales472

ⅩⅢ.1.2 Gautschi's and Deuflhard's Trigonometric Methods473

ⅩⅢ.1.3 The Impulse Method475

ⅪⅢ.1.4 The Mollified Impulse Method476

ⅩⅢ.1.5 Gautschi's Method Revisited477

ⅩⅢ.1.6 Two-Force Methods478

ⅩⅢ.2 A Nonlinear Model Problem and Numerical Phenomena478

ⅩⅢ.2.1 Time Scales in the Fermi-Pasta-Ulam Problem479

ⅩⅢ.2.2 Numerical Methods481

ⅩⅢ.2.3 Accuracy Comparisons482

ⅩⅢ.2.4 Energy Exchange between Stiff Components483

ⅩⅢ.2.5 Near-Conservation of Total and Oscillatory Energy484

ⅩⅢ.3 Principal Terms of the Modulated Fourier Expansion486

ⅩⅢ.3.1 Decomposition of the Exact Solution486

ⅩⅢ.3.2 Decomposition of the Numerical Solution488

ⅩⅢ.4 Accuracy and Slow Exchange490

ⅩⅢ.4.1 Convergence Properties on Bounded Time Intervals490

ⅩⅢ.4.2 Intra-Oscillatory and Oscillatory-Smooth Exchanges494

ⅩⅢ.5 Modulated Fourier Expansions496

ⅩⅢ.5.1 Expansion of the Exact Solution496

ⅩⅢ.5.2 Expansion of the Numerical Solution498

ⅩⅢ.5.3 Expansion of the Velocity Approximation502

ⅩⅢ.6 Almost-Invariants of the Modulated Fourier Expansions503

ⅪⅢ.6.1 The Hamiltonian of the Modulated Fourier Expansion503

ⅩⅢ.6.2 A Formal Invariant Close to the Oscillatory Energy505

ⅩⅢ.6.3 Almost-Invariants of the Numerical Method507

ⅩⅢ.7 Long-Time Near-Conservation of Total and Oscillatory Energy510

ⅩⅢ.8 Energy Behaviour of the St？rmer-Verlet Method513

ⅩⅢ.9 Systems with Several Constant Frequencies516

ⅩⅢ.9.1 Oscillatory Energies and Resonances517

ⅩⅢ.9.2 Multi-Frequency Modulated Fourier Expansions519

ⅩⅢ.9.3 Almost-Invariants of the Modulation System521

ⅩⅢ.9.4 Long-Time Near-Conservation of Total and Oscillatory Energies524

ⅩⅢ.10 Systems with Non-Constant Mass Matrix526

ⅩⅢ.11 Exercises529

Ⅹ．Oscillatory Differential Equations with Varying High Frequencies531

ⅩⅣ.1 Linear Systems with Time-Dependent Skew-Hermitian Matrix531

ⅩⅣ.1.1 Adiabatic Transformation and Adiabatic Invariants531

ⅩⅣ.1.2 Adiabatic Integrators536

ⅩⅣ.2 Mechanical Systems with Time-Dependent Frequencies539

ⅩⅣ.2.1 Canonical Transformation to Adiabatic Variables540

ⅩⅣ.2.2 Adiabatic Integrators547

ⅩⅣ.2.3 Error Analysis of the Impulse Method550

ⅩⅣ.2.4 Error Analysis of the Mollified Impulse Method554

ⅩⅣ.3 Mechanical Systems with Solution-Dependent Frequencies555

ⅩⅣ.3.1 Constraining Potentials555

ⅩⅣ.3.2 Transformation to Adiabatic Variables558

ⅩⅣ.3.3 Integrators in Adiabatic Variables563

ⅩⅣ.3.4 Analysis of Multiple Time-Stepping Methods564

ⅩⅣ.4 Exercises564

ⅩⅤ．Dynamics of Multistep Methods567

ⅩⅤ.1 Numerical Methods and Experiments567

ⅩⅤ.1.1 Linear Multistep Methods567

ⅩⅤ.1.2 Multistep Methods for Second Order Equations569

ⅩⅤ.1.3 Partitioned Multistep Methods572

ⅩⅤ.2 The Underlying One-Step Method573

ⅩⅤ.2.1 Strictly Stable Multistep methods573

ⅩⅤ.2.2 Formal Analysis for Weakly Stable Methods575

ⅩⅤ.3 Backward Error Analysis576

ⅩⅤ.3.1 Modified Equation for Smooth Numerical Solutions576

ⅩⅤ.3.2 Parasitic Modified Equations579

ⅩⅤ.4 Can Multistep Methods be Symplectic？585

ⅩⅤ.4.1 Non-Symplecticity of the Underlying One-Step Method585

ⅩⅤ.4.2 Symplecticity in the Higher-Dimensional Phase Space587

ⅩⅤ.4.3 Modified Hamiltonian of Multistep Methods589

ⅩⅤ.4.4 Modified Quadratic First Integrals591

ⅩⅤ.5 Long-Term Stability592

ⅩⅤ.5.1 Role of Growth Parameters592

ⅩⅤ.5.2 Hamiltonian of the Full Modified System594

ⅩⅤ.5.3 Long-Time Bounds for Parasitic Solution Components596

ⅩⅤ.6 Explanation of the Long-Time Behaviour600

ⅩⅤ.6.1 Conservation of Energy and Angular Momentum600

ⅩⅤ.6.2 Linear Error Growth for Integrable Systems601

ⅩⅤ.7 Practical Considerations602

ⅩⅤ.7.1 Numerical Instabilities and Resonances602

ⅩⅤ.7.2 Extension to Variable Step Sizes605

ⅩⅤ.8 Multi-Value or General Linear Methods609

ⅩⅤ.8.1 Underlying One-Step Method and Backward Error Analysis609

ⅩⅤ.8.2 Symplecticity and Symmetry611

ⅩⅤ.8.3 Growth Parameters614

ⅩⅤ.9 Exercises615

Bibliography617

Index637