Geometric Numerical Integration : Structure-Preserving Algorithms for Ordinary Differential Equations

Structure-Preserving Algorithms for Ordinary Differential Equations

By: Christian Lubich, Gerhard Wanner, Ernst Hairer

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Hardcover | 1 April 2006 | Edition Number 2

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Hardcover
664 Pages

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Revised

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24.0 x 16.2 x 4.1

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Numerical methods that preserve properties of Hamiltonian systems, reversible systems, differential equations on manifolds and problems with highly oscillatory solutions are the subject of this book. A complete self-contained theory of symplectic and symmetric methods, which include Runge-Kutta, composition, splitting, multistep and various specially designed integrators, is presented and their construction and practical merits are discussed. The long-time behaviour of the numerical solutions is studied using a backward error analysis (modified equations) combined with KAM theory. The book is illustrated by many figures, it treats applications from physics and astronomy and contains many numerical experiments and comparisons of different approaches. The second edition is substantially revised and enlarged, with many improvements in the presentation and additions concerning in particular non-canonical Hamiltonian systems, highly oscillatory mechanical systems, and the dynamics of multistep methods.

Industry Reviews

From the reviews of the second edition:

"This book is highly recommended for advanced courses in numerical methods for ordinary differential equations as well as a reference for researchers/developers in the field of geometric integration, differential equations in general and related subjects. It is a must for academic and industrial libraries." -- MATHEMATICAL REVIEWS

"The second revised edition of the monograph is a fine work organized in fifteen chapters, updated and extended. ... The material of the book is organized in sections which are ... self-contained, so that one can dip into the book to learn a particular topic ... . A person interested in geometrical numerical integration will find this book extremely useful." (Calin Ioan Gheorghiu, Zentralblatt MATH, Vol. 1094 (20), 2006)

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Non-Fiction Mathematics Calculus & Mathematical Analysis Numerical Analysis Differential Calculus & Equations Science Physics Applied Mathematics

Examples and Numerical Experiments	p. 1
First Problems and Methods	p. 1
The Lotka-Volterra Model	p. 1
First Numerical Methods	p. 3
The Pendulum as a Hamiltonian System	p. 4
The Stormer-Verlet Scheme	p. 7
The Kepler Problem and the Outer Solar System	p. 8
Angular Momentum and Kepler's Second Law	p. 9
Exact Integration of the Kepler Problem	p. 10
Numerical Integration of the Kepler Problem	p. 12
The Outer Solar System	p. 13
The Henon-Heiles Model	p. 15
Molecular Dynamics	p. 18
Highly Oscillatory Problems	p. 21
A Fermi-Pasta-Ulam Problem	p. 21
Application of Classical Integrators	p. 23
Exercises	p. 24
Numerical Integrators	p. 27
Runge-Kutta and Collocation Methods	p. 27
Runge-Kutta Methods	p. 28
Collocation Methods	p. 30
Gauss and Lobatto Collocation	p. 34
Discontinuous Collocation Methods	p. 35
Partitioned Runge-Kutta Methods	p. 38
Definition and First Examples	p. 38
Lobatto IIIA-IIIB Pairs	p. 40
Nystrom Methods	p. 41
The Adjoint of a Method	p. 42
Composition Methods	p. 43
Splitting Methods	p. 47
Exercises	p. 50
Order Conditions, Trees and B-Series	p. 51
Runge-Kutta Order Conditions and B-Series	p. 51
Derivation of the Order Conditions	p. 51
B-Series	p. 56
Composition of Methods	p. 59
Composition of B-Series	p. 61
The Butcher Group	p. 64
Order Conditions for Partitioned Runge-Kutta Methods	p. 66
Bi-Coloured Trees and P-Series	p. 66
Order Conditions for Partitioned Runge-Kutta Methods	p. 68
Order Conditions for Nystrom Methods	p. 69
Order Conditions for Composition Methods	p. 71
Introduction	p. 71
The General Case	p. 73
Reduction of the Order Conditions	p. 75
Order Conditions for Splitting Methods	p. 80
The Baker-Campbell-Hausdorff Formula	p. 83
Derivative of the Exponential and Its Inverse	p. 83
The BCH Formula	p. 84
Order Conditions via the BCH Formula	p. 87
Calculus of Lie Derivatives	p. 87
Lie Brackets and Commutativity	p. 89
Splitting Methods	p. 91
Composition Methods	p. 92
Exercises	p. 95
Conservation of First Integrals and Methods on Manifolds	p. 97
Examples of First Integrals	p. 97
Quadratic Invariants	p. 101
Runge-Kutta Methods	p. 101
Partitioned Runge-Kutta Methods	p. 102
Nystrom Methods	p. 104
Polynomial Invariants	p. 105
The Determinant as a First Integral	p. 105
Isospectral Flows	p. 107
Projection Methods	p. 109
Numerical Methods Based on Local Coordinates	p. 113
Manifolds and the Tangent Space	p. 114
Differential Equations on Manifolds	p. 115
Numerical Integrators on Manifolds	p. 116
Differential Equations on Lie Groups	p. 118
Methods Based on the Magnus Series Expansion	p. 121
Lie Group Methods	p. 123
Crouch-Grossman Methods	p. 124
Munthe-Kaas Methods	p. 125
Further Coordinate Mappings	p. 128
Geometric Numerical Integration Meets Geometric Numerical Linear Algebra	p. 131
Numerical Integration on the Stiefel Manifold	p. 131
Differential Equations on the Grassmann Manifold	p. 135
Dynamical Low-Rank Approximation	p. 137
Exercises	p. 139
Symmetric Integration and Reversibility	p. 143
Reversible Differential Equations and Maps	p. 143
Symmetric Runge-Kutta Methods	p. 146
Collocation and Runge-Kutta Methods	p. 146
Partitioned Runge-Kutta Methods	p. 148
Symmetric Composition Methods	p. 149
Symmetric Composition of First Order Methods	p. 150
Symmetric Composition of Symmetric Methods	p. 154
Effective Order and Processing Methods	p. 158
Symmetric Methods on Manifolds	p. 161
Symmetric Projection	p. 161
Symmetric Methods Based on Local Coordinates	p. 166
Energy - Momentum Methods and Discrete Gradients	p. 171
Exercises	p. 176
Symplectic Integration of Hamiltonian Systems	p. 179
Hamiltonian Systems	p. 180
Lagrange's Equations	p. 180
Hamilton's Canonical Equations	p. 181
Symplectic Transformations	p. 182
First Examples of Symplectic Integrators	p. 187
Symplectic Runge-Kutta Methods	p. 191
Criterion of Symplecticity	p. 191
Connection Between Symplectic and Symmetric Methods	p. 194
Generating Functions	p. 195
Existence of Generating Functions	p. 195
Generating Function for Symplectic Runge-Kutta Methods	p. 198
The Hamilton-Jacobi Partial Differential Equation	p. 200
Methods Based on Generating Functions	p. 203
Variational Integrators	p. 204
Hamilton's Principle	p. 204
Discretization of Hamilton's Principle	p. 206
Symplectic Partitioned Runge-Kutta Methods Revisited	p. 208
Noether's Theorem	p. 210
Hamiltonian Perturbation Theory and Symplectic Integrators	p. 389
Completely Integrable Hamiltonian Systems	p. 390
Local Integration by Quadrature	p. 390
Completely Integrable Systems	p. 393
Action-Angle Variables	p. 397
Conditionally Periodic Flows	p. 399
The Toda Lattice - an Integrable System	p. 402
Transformations in the Perturbation Theory for Integrable Systems	p. 404
The Basic Scheme of Classical Perturbation Theory	p. 405
Lindstedt-Poincare Series	p. 406
Kolmogorov's Iteration	p. 410
Birkhoff Normalization Near an Invariant Torus	p. 412
Linear Error Growth and Near-Preservation of First Integrals	p. 413
Near-Invariant Tori on Exponentially Long Times	p. 417
Estimates of Perturbation Series	p. 417
Near-Invariant Tori of Perturbed Integrable Systems	p. 421
Near-Invariant Tori of Symplectic Integrators	p. 422
Kolmogorov's Theorem on Invariant Tori	p. 423
Kolmogorov's Theorem	p. 423
KAM Tori under Symplectic Discretization	p. 428
Invariant Tori of Symplectic Maps	p. 430
A KAM Theorem for Symplectic Near-Identity Maps	p. 431
Invariant Tori of Symplectic Integrators	p. 433
Strongly Non-Resonant Step Sizes	p. 433
Exercises	p. 434
Reversible Perturbation Theory and Symmetric Integrators	p. 437
Integrable Reversible Systems	p. 437
Transformations in Reversible Perturbation Theory	p. 442
The Basic Scheme of Reversible Perturbation Theory	p. 443
Reversible Perturbation Series	p. 444
Reversible KAM Theory	p. 445
Reversible Birkhoff-Type Normalization	p. 447
Linear Error Growth and Near-Preservation of First Integrals	p. 448
Invariant Tori under Reversible Discretization	p. 451
Near-Invariant Tori over Exponentially Long Times	p. 451
A KAM Theorem for Reversible Near-Identity Maps	p. 451
Exercises	p. 453
Dissipatively Perturbed Hamiltonian and Reversible Systems	p. 455
Numerical Experiments with Van der Pol's Equation	p. 455
Averaging Transformations	p. 458
The Basic Scheme of Averaging	p. 458
Perturbation Series	p. 459
Attractive Invariant Manifolds	p. 460
Weakly Attractive Invariant Tori of Perturbed Integrable Systems	p. 464
Weakly Attractive Invariant Tori of Numerical Integrators	p. 465
Modified Equations of Perturbed Differential Equations	p. 466
Symplectic Methods	p. 467
Symmetric Methods	p. 469
Exercises	p. 469
Oscillatory Differential Equations with Constant High Frequencies	p. 471
Towards Longer Time Steps in Solving Oscillatory Equations of Motion	p. 471
The Stormer-Verlet Method vs. Multiple Time Scales	p. 472
Gautschi's and Deuflhard's Trigonometric Methods	p. 473
The Impulse Method	p. 475
The Mollified Impulse Method	p. 476
Gautschi's Method Revisited	p. 477
Two-Force Methods	p. 478
A Nonlinear Model Problem and Numerical Phenomena	p. 478
Time Scales in the Fermi-Pasta-Ulam Problem	p. 479
Numerical Methods	p. 481
Accuracy Comparisons	p. 482
Energy Exchange between Stiff Components	p. 483
Near-Conservation of Total and Oscillatory Energy	p. 484
Principal Terms of the Modulated Fourier Expansion	p. 486
Decomposition of the Exact Solution	p. 486
Decomposition of the Numerical Solution	p. 488
Accuracy and Slow Exchange	p. 490
Convergence Properties on Bounded Time Intervals	p. 490
Intra-Oscillatory and Oscillatory-Smooth Exchanges	p. 494
Modulated Fourier Expansions	p. 496
Expansion of the Exact Solution	p. 496
Expansion of the Numerical Solution	p. 498
Expansion of the Velocity Approximation	p. 502
Almost-Invariants of the Modulated Fourier Expansions	p. 503
The Hamiltonian of the Modulated Fourier Expansion	p. 503
A Formal Invariant Close to the Oscillatory Energy	p. 505
Almost-Invariants of the Numerical Method	p. 507
Long-Time Near-Conservation of Total and Oscillatory Energy	p. 510
Energy Behaviour of the Stormer-Veriet Method	p. 513
Systems with Several Constant Frequencies	p. 516
Oscillatory Energies and Resonances	p. 517
Multi-Frequency Modulated Fourier Expansions	p. 519
Almost-Invariants of the Modulation System	p. 521
Long-Time Near-Conservation of Total and Oscillatory Energies	p. 524
Systems with Non-Constant Mass Matrix	p. 526
Exercises	p. 529
Oscillatory Differential Equations with Varying High Frequencies	p. 531
Linear Systems with Time-Dependent Skew-Hermitian Matrix	p. 531
Adiabatic Transformation and Adiabatic Invariants	p. 531
Adiabatic Integrators	p. 536
Mechanical Systems with Time-Dependent Frequencies	p. 539
Canonical Transformation to Adiabatic Variables	p. 540
Adiabatic Integrators	p. 547
Error Analysis of the Impulse Method	p. 550
Error Analysis of the Mollified Impulse Method	p. 554
Mechanical Systems with Solution-Dependent Frequencies	p. 555
Constraining Potentials	p. 555
Transformation to Adiabatic Variables	p. 558
Integrators in Adiabatic Variables	p. 563
Analysis of Multiple Time-Stepping Methods	p. 564
Exercises	p. 564
Dynamics of Multistep Methods	p. 567
Numerical Methods and Experiments	p. 567
Linear Multistep Methods	p. 567
Multistep Methods for Second Order Equations	p. 569
Partitioned Multistep Methods	p. 572
The Underlying One-Step Method	p. 573
Strictly Stable Multistep methods	p. 573
Formal Analysis for Weakly Stable Methods	p. 575
Backward Error Analysis	p. 576
Modified Equation for Smooth Numerical Solutions	p. 576
Parasitic Modified Equations	p. 579
Can Multistep Methods be Symplectic?	p. 585
Non-Symplecticity of the Underlying One-Step Method	p. 585
Symplecticity in the Higher-Dimensional Phase Space	p. 587
Modified Hamiltonian of Multistep Methods	p. 589
Modified Quadratic First Integrals	p. 591
Long-Term Stability	p. 592
Role of Growth Parameters	p. 592
Hamiltonian of the Full Modified System	p. 594
Long-Time Bounds for Parasitic Solution Components	p. 596
Explanation of the Long-Time Behaviour	p. 600
Conservation of Energy and Angular Momentum	p. 600
Linear Error Growth for Integrable Systems	p. 601
Practical Considerations	p. 602
Numerical Instabilities and Resonances	p. 602
Extension to Variable Step Sizes	p. 605
Multi-Value or General Linear Methods	p. 609
Underlying One-Step Method and Backward Error Analysis	p. 609
Symplecticity and Symmetry	p. 611
Growth Parameters	p. 614
Exercises	p. 615
Bibliography	p. 617
Index	p. 637
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Geometric Numerical Integration : Structure-Preserving Algorithms for Ordinary Differential Equations

Structure-Preserving Algorithms for Ordinary Differential Equations

At a Glance

Hardcover

Industry Reviews

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