Geometric numerical integration: structure-preserving algorithms for ordinary differential equations
Gespeichert in:
| Hauptverfasser: | , , |
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| Format: | Elektronisch E-Book |
| Sprache: | English |
| Veröffentlicht: |
Berlin [u.a.]
Springer
2006
|
| Ausgabe: | 2. ed. |
| Schriftenreihe: | Springer series in computational mathematics
31 |
| Schlagworte: | |
| Online-Zugang: | BTU01 TUM01 UBA01 UBM01 UBR01 UBT01 UPA01 Volltext Inhaltsverzeichnis |
| Beschreibung: | 1 Online-Ressource (XVII, 644 S.) Ill., graph. Darst. |
| ISBN: | 9783540306665 |
| DOI: | 10.1007/3-540-30666-8 |
Internformat
MARC
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| 100 | 1 | |a Hairer, Ernst |d 1949- |e Verfasser |0 (DE-588)139445188 |4 aut | |
| 245 | 1 | 0 | |a Geometric numerical integration |b structure-preserving algorithms for ordinary differential equations |c Ernst Hairer ; Christian Lubich ; Gerhard Wanner |
| 250 | |a 2. ed. | ||
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| 700 | 1 | |a Wanner, Gerhard |d 1942- |e Verfasser |0 (DE-588)13944534X |4 aut | |
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Datensatz im Suchindex
| _version_ | 1804136322550464512 |
|---|---|
| adam_text | I. Examples and Numerical Experiments 1
1.1 First Problems and Methods 1
1.1.1 The Lotka Volterra Model 1
1.1.2 First Numerical Methods 3
1.1.3 The Pendulum as a Hamiltonian System 4
1.1.4 The Störmer Verlet Scheme 7
1.2 The Kepler Problem and the Outer Solar System 8
1.2.1 Angular Momentum and Kepler s Second Law 9
1.2.2 Exact Integration of the Kepler Problem 10
1.2.3 Numerical Integration of the Kepler Problem 12
1.2.4 The Outer Solar System 13
1.3 The Henon Heiles Model 15
1.4 Molecular Dynamics 18
1.5 Highly Oscillatory Problems 21
1.5.1 A Fermi Pasta Ulam Problem 21
1.5.2 Application of Classical Integrators 23
1.6 Exercises 24
II. Numerical Integrators 27
II. 1 Runge Kutta and Collocation Methods 27
II. 1.1 Runge Kutta Methods 28
II. 1.2 Collocation Methods 30
II. 1.3 Gauss and Lobatto Collocation 34
II. 1.4 Discontinuous Collocation Methods 35
11.2 Partitioned Runge Kutta Methods 38
11.2.1 Definition and First Examples 38
11.2.2 Lobatto IIIA IIIB Pairs 40
11.2.3 Nyström Methods 41
11.3 TheAdjointofaMethod 42
11.4 Composition Methods 43
11.5 Splitting Methods 47
11.6 Exercises 50
III. Order Conditions, Trees and B Series 51
III. 1 Runge Kutta Order Conditions and B Series 51
III. 1.1 Derivation of the Order Conditions 51
III. 1.2 B Series 56
III. 1.3 Composition of Methods 59
III. 1.4 Composition of B Series 61
III. 1.5 The Butcher Group 64
111.2 Order Conditions for Partitioned Runge Kutta Methods 66
111.2.1 Bi Coloured Trees and P Series 66
111.2.2 Order Conditions for Partitioned Runge Kutta Methods 68
111.2.3 Order Conditions for Nyström Methods 69
111.3 Order Conditions for Composition Methods 71
111.3.1 Introduction 71
111.3.2 The General Case 73
111.3.3 Reduction of the Order Conditions 75
111.3.4 Order Conditions for Splitting Methods 80
111.4 The Baker Campbell Hausdorff Formula 83
111.4.1 Derivative of the Exponential and Its Inverse 83
111.4.2 The BCH Formula 84
111.5 Order Conditions via the BCH Formula 87
111.5.1 Calculus of Lie Derivatives 87
111.5.2 Lie Brackets and Commutativity 89
111.5.3 Splitting Methods 91
111.5.4 Composition Methods 92
111.6 Exercises 95
IV. Conservation of First Integrals and Methods on Manifolds 97
IV. 1 Examples of First Integrals 97
IV.2 Quadratic Invariants 101
IV.2.1 Runge Kutta Methods 101
IV.2.2 Partitioned Runge Kutta Methods 102
IV.2.3 Nyström Methods 104
IV.3 Polynomial Invariants 105
IV.3.1 The Determinant as a First Integral 105
IV.3.2 Isospectral Flows 107
IV.4 Projection Methods 109
IV.5 Numerical Methods Based on Local Coordinates 113
IV.5.1 Manifolds and the Tangent Space 114
IV.5.2 Differential Equations on Manifolds 115
IV.5.3 Numerical Integrators on Manifolds 116
IV.6 Differential Equations on Lie Groups 118
IV.7 Methods Based on the Magnus Series Expansion 121
IV.8 Lie Group Methods 123
IV.8.1 Crouch Grossman Methods 124
IV.8.2 Munthe Kaas Methods 125
IV.8.3 Further Coordinate Mappings 128
IV.9 Geometrie Numerical Integration Meets Geometrie Numerical
Linear Algebra 131
IV.9.1 Numerical Integration on the Stiefel Manifold 131
IV.9.2 Differential Equations on the Grassmann Manifold .... 135
IV.9.3 Dynamical Low Rank Approximation 137
IV.10 Exercises 139
V. Symmetrie Integration and Reversibility 143
VI Reversible Differential Equations and Maps 143
V.2 Symmetrie Runge Kutta Methods 146
V.2.1 Collocation and Runge Kutta Methods 146
V.2.2 Partitioned Runge Kutta Methods 148
V.3 Symmetrie Composition Methods 149
V.3.1 Symmetrie Composition of First Order Methods 150
V.3.2 Symmetrie Composition of Symmetrie Methods 154
V.3.3 Effective Order and Processing Methods 158
V.4 Symmetrie Methods on Manifolds 161
V.4.1 Symmetrie Projection 161
V.4.2 Symmetrie Methods Based on Local Coordinates 166
V.5 Energy Momentum Methods and Discrete Gradients 171
V.6 Exercises 176
VI. Symplectic Integration of Hamiltonian Systems 179
VI.l Hamiltonian Systems 180
VI. 1.1 Lagrange s Equations 180
VI. 1.2 Hamilton s Canonical Equations 181
VI.2 Symplectic Transformations 182
VI.3 First Examples of Symplectic Integrators 187
VI.4 Symplectic Runge Kutta Methods 191
VI.4.1 Criterion of Symplecticity 191
VI.4.2 Connection Between Symplectic and Symmetrie
Methods 194
VI.5 Generating Functions 195
VI.5.1 Existence of Generating Functions 195
VI.5.2 Generating Function for Symplectic Runge Kutta
Methods 198
VI.5.3 The Hamilton Jacobi Partial Differential Equation 200
VI.5.4 Methods Based on Generating Functions 203
VI.6 Variational Integrators 204
VI.6.1 Hamilton s Principle 204
VI.6.2 Discretization of Hamilton s Principle 206
VI.6.3 Symplectic Partitioned Runge Kutta Methods
Revisited 208
VI.6.4 Noether s Theorem 210
VI.7 Characterization of Symplectic Methods 212
VI.7.1 B Series Methods Conserving Quadratic First Integrals 212
VI.7.2 Characterization of Symplectic P Series (and B Series) 217
VI.7.3 Irreducible Runge Kutta Methods 220
VI.7.4 Characterization of Irreducible Symplectic Methods ... 222
VI.8 Conjugate Symplecticity 222
VI.8.1 Examples and Order Conditions 223
VI.8.2 Near Conservation of Quadratic First Integrals 225
VI.9 Volume Preservation 227
VI.10 Exercises 233
VII. Non Canonical Hamiltonian Systems 237
VII. 1 Constrained Mechanical Systems 237
VII. 1.1 Introduction and Examples 237
VII. 1.2 Hamiltonian Formulation 239
VII. 1.3 A Symplectic First Order Method 242
VII. 1.4 SHAKE and RATTLE 245
VII. 1.5 The Lobatto IIIA IIIB Pair 247
VII. 1.6 Splitting Methods 252
VII.2 Poisson Systems 254
VII.2.1 Canonical Poisson Structure 254
VII.2.2 General Poisson Structures 256
VII.2.3 Hamiltonian Systems on Symplectic Submanifolds.... 258
VII.3 The Darboux Lie Theorem 261
VII.3.1 Commutativity of Poisson Flows and Lie Brackets .... 261
VII.3.2 Simultaneous Linear Partial Differential Equations .... 262
VII.3.3 Coordinate Changes and the Darboux Lie Theorem ... 265
VII.4 Poisson Integrators 268
VII.4.1 Poisson Maps and Symplectic Maps 268
VII.4.2 Poisson Integrators 270
VII.4.3 Integrators Based on the Darboux Lie Theorem 272
VII.5 Rigid Body Dynamics and Lie Poisson Systems 274
VII.5.1 History of the Euler Equations 275
VII.5.2 Hamiltonian Formulation of Rigid Body Motion 278
VII.5.3 Rigid Body Integrators 280
VII.5.4 Lie Poisson Systems 286
VII.5.5 Lie Poisson Reduction 289
VII.6 Reduced Models of Quantum Dynamics 293
VII.6.1 Hamiltonian Structure of the Schrödinger Equation ... 293
VII.6.2 The Dirac Frenkel Variational Principle 295
VII.6.3 Gaussian Wavepacket Dynamics 296
VII.6.4 A Splitting Integrator for Gaussian Wavepackets 298
VII.7 Exercises 301
VIII. Structure Preserving Implementation 303
VIII. 1 Dangers of Using Standard Step Size Control 303
VIII.2 Time Transformations 306
VIII.2.1 Symplectic Integration 306
VIII.2.2 Reversible Integration 309
VIII.3 Structure Preserving Step Size Control 310
VIII.3.1 Proportional, Reversible Controllers 310
VIII.3.2 Integrating, Reversible Controllers 314
VIII.4 Multiple Time Stepping 316
VIII.4.1 Fast Slow Splitting: the Impulse Method 317
VIII.4.2 Averaged Forces 319
VIII.5 Reducing Rounding Errors 322
VIII.6 Implementation of Implicit Methods 325
VIII.6.1 Starting Approximations 326
VIII.6.2 Fixed Point Versus Newton Iteration 330
VIII.7 Exercises 335
IX. Backward Error Analysis and Structure Preservation 337
IX. 1 Modified Differential Equation Examples 337
IX.2 Modified Equations of Symmetrie Methods 342
IX. 3 Modified Equations of Symplectic Methods 343
IX.3.1 Existence of a Local Modified Hamiltonian 343
IX.3.2 Existence of a Global Modified Hamiltonian 344
IX.3.3 Poisson Integrators 347
IX.4 Modified Equations of Splitting Methods 348
IX.5 Modified Equations of Methods on Manifolds 350
IX.5.1 Methods on Manifolds and First Integrals 350
IX.5.2 Constrained Hamiltonian Systems 352
IX.5.3 Lie Poisson Integrators 354
IX.6 Modified Equations for Variable Step Sizes 356
IX.7 Rigorous Estimates Local Error 358
IX.7.1 Estimation of the Derivatives of the Numerical Solution 360
IX.7.2 Estimation of the Coefficients of the Modified Equation 362
IX.7.3 Choice of TV and the Estimation of the Local Error 364
IX.8 Long Time Energy Conservation 366
IX.9 Modified Equation in Terms of Trees 369
IX.9.1 B Series of the Modified Equation 369
IX.9.2 Elementary Hamiltonians 373
IX.9.3 Modified Hamiltonian 375
IX.9.4 First Integrals Close to the Hamiltonian 375
IX.9.5 Energy Conservation: Examples and Counter Examples 379
IX.10 Extension to Partitioned Systems 381
IX.10.1 P Series of the Modified Equation 381
IX.10.2 Elementary Hamiltonians 384
IX.ll Exercises 386
X. Hamiltonian Perturbation Theory and Symplectic Integrators 389
X. 1 Completely Integrable Hamiltonian Systems 390
X. 1.1 Local Integration by Quadrature 390
X.1.2 Completely Integrable Systems 393
X.1.3 Action Angle Variables 397
X.1.4 Conditionally Periodic Flows 399
X. 1.5 The Toda Lattice an Integrable System 402
X.2 Transformations in the Perturbation Theory for Integrable
Systems 404
X.2.1 The Basic Scheme of Classical Perturbation Theory ... 405
X.2.2 Lindstedt Poincare Series 406
X.2.3 Kolmogorov s Iteration 410
X.2.4 Birkhoff Normalization Near an Invariant Torus 412
X.3 Linear Error Growth and Near Preservation of First Integrals ... 413
X.4 Near Invariant Tori on Exponentially Long Times 417
X.4.1 Estimates of Perturbation Series 417
X.4.2 Near Invariant Tori of Perturbed Integrable Systems ... 421
X.4.3 Near Invariant Tori of Symplectic Integrators 422
X.5 Kolmogorov s Theorem on Invariant Tori 423
X.5.1 Kolmogorov s Theorem 423
X.5.2 KAM Tori under Symplectic Discretization 428
X.6 Invariant Tori of Symplectic Maps 430
X.6.1 A KAM Theorem for Symplectic Near Identity Maps . 431
X.6.2 Invariant Tori of Symplectic Integrators 433
X.6.3 Strongly Non Resonant Step Sizes 433
X.7 Exercises 434
XI. Reversible Perturbation Theory and Symmetrie Integrators 437
XI.l Integrable Reversible Systems 437
XI.2 Transformations in Reversible Perturbation Theory 442
XI.2.1 The Basic Scheme of Reversible Perturbation Theory.. 443
XI.2.2 Reversible Perturbation Series 444
XI.2.3 Reversible KAM Theory 445
XI.2.4 Reversible Birkhoff Type Normalization 447
XI.3 Linear Error Growth and Near Preservation of First Integrals ... 448
XI.4 Invariant Tori under Reversible Discretization 451
XI.4.1 Near Invariant Tori over Exponentially Long Times ... 451
XI.4.2 A KAM Theorem for Reversible Near Identity Maps .. 451
XI.5 Exercises 453
XII. Dissipatively Perturbed Hamiltonian and Reversible Systems 455
XII. 1 Numerical Experiments with Van der Pol s Equation 455
XII.2 Averaging Transformations 458
XII.2.1 The Basic Scheme of Averaging 458
XII.2.2 Perturbation Series 459
XII.3 Attractive Invariant Manifolds 460
XII.4 Weakly Attractive Invariant Tori of Perturbed Integrable Systems 464
XII.5 Weakly Attractive Invariant Tori of Numerical Integrators 465
XII.5.1 Modified Equations of Perturbed Differential Equations 466
XII.5.2 Symplectic Methods 467
XII.5.3 Symmetrie Methods 469
XII.6 Exercises 469
XIII. Oscillatory Differential Equations with Constant High Frequencies. 471
XIII. 1 Towards Longer Time Steps in Solving Oscillatory Equations
of Motion 471
XIII. 1.1 The Stornier Verlet Method vs. Multiple Time Scales .472
XIII. 1.2 Gautschi s and Deuflhard s Trigonometrie Methods ... 473
XIII.1.3 The Impulse Method 475
XIII.1.4 The Mollified Impulse Method 476
XIII. 1.5 Gautschi s Method Revisited 477
XIII.1.6 Two Force Methods 478
XIII.2 A Nonlinear Model Problem and Numerical Phenomena 478
XIII.2.1 Time Scales in the Fermi Pasta Ulam Problem 479
XIII.2.2 Numerical Methods 481
XIII.2.3 Accuracy Comparisons 482
XIII.2.4 Energy Exchange between Stiff Components 483
XIII.2.5 Near Conservation of Total and Oscillatory Energy.... 484
XIII.3 Principal Terms of the Modulated Fourier Expansion 486
XIII.3.1 Decomposition of the Exact Solution 486
XIII.3.2 Decomposition of the Numerical Solution 488
XIII.4 Accuracy and Slow Exchange 490
XIII.4.1 Convergence Properties on Bounded Time Intervals ... 490
XIII.4.2 Intra Oscillatory and Oscillatory Smooth Exchanges .. 494
XIII.5 Modulated Fourier Expansions 496
XIII.5.1 Expansion of the Exact Solution 496
XIII.5.2 Expansion of the Numerical Solution 498
XIII.5.3 Expansion of the Velocity Approximation 502
XIII.6 Almost Invariants of the Modulated Fourier Expansions 503
XIII.6.1 The Hamiltonian of the Modulated Fourier Expansion . 503
XIII.6.2 A Formal Invariant Close to the Oscillatory Energy ... 505
XIII.6.3 Almost Invariants of the Numerical Method 507
XIII.7 Long Time Near Conservation of Total and Oscillatory Energy .510
XIII.8 Energy Behaviour of the Störmer Verlet Method 513
XIII.9 Systems with Several Constant Frequencies 516
XIII.9.1 Oscillatory Energies and Resonances 517
XIII.9.2 Multi Frequency Modulated Fourier Expansions 519
XIII.9.3 Almost Invariants of the Modulation System 521
XIII.9.4 Long Time Near Conservation of Total and
Oscillatory Energies 524
XIII. 10 Systems with Non Constant Mass Matrix 526
XIII.ll Exercises 529
XIV. Oscülatory Differential Equations with Varying High Frequencies .531
XIV. 1 Linear Systems with Time Dependent Skew Hermitian Matrix ..531
XIV. 1.1 Adiabatic Transformation and Adiabatic Invariants .... 531
XIV.1.2 Adiabatic Integrators 536
XIV.2 Mechanical Systems with Time Dependent Frequencies 539
XIV.2.1 Canonical Transformation to Adiabatic Variables 540
XIV.2.2 Adiabatic Integrators 547
XIV.2.3 Error Analysis of the Impulse Method 550
XIV.2.4 Error Analysis of the Mollified Impulse Method 554
XIV.3 Mechanical Systems with Solution Dependent Frequencies 555
XIV.3.1 Constraining Potentials 555
XIV.3.2 Transformation to Adiabatic Variables 558
XIV.3.3 Integrators in Adiabatic Variables 563
XIV.3.4 Analysis of Multiple Time Stepping Methods 564
XIV.4 Exercises 564
XV. Dynamics of Multistep Methods 567
XV. 1 Numerical Methods and Experiments 567
XV.1.1 Linear Multistep Methods 567
XV. 1.2 Multistep Methods for Second Order Equations 569
XV. 1.3 Partitioned Multistep Methods 572
XV.2 The Underlying One Step Method 573
XV.2.1 Strictly Stable Multistep methods 573
XV.2.2 Formal Analysis for Weakly Stable Methods 575
XV.3 Backward Error Analysis 576
XV.3.1 Modified Equation for Smooth Numerical Solutions ... 576
XV.3.2 Parasitic Modified Equations 579
XV.4 Can Multistep Methods be Symplectic? 585
XV.4.1 Non Symplecticity of the Underlying One Step Method 585
XV.4.2 Symplecticity in the Higher Dimensional Phase Space . 587
XV.4.3 Modified Hamiltonian of Multistep Methods 589
XV.4.4 Modified Quadratic First Integrals 591
XV.5 Long Term Stability 592
XV.5.1 Role of Growth Parameters 592
XV.5.2 Hamiltonian of the Füll Modified System 594
XV.5.3 Long Time Bounds for Parasitic Solution Components 596
XV.6 Explanation of the Long Time Behaviour 600
XV.6.1 Conservation of Energy and Angular Momentum 600
XV.6.2 Linear Error Growth for Integrable Systems 601
XV.7 Practical Considerations 602
XV.7.1 Numerical Instabilities and Resonances 602
XV.7.2 Extension to Variable Step Sizes 605
XV. 8 Multi Value or General Linear Methods 609
XV.8.1 Underlying One Step Method and Backward Error
Analysis 609
XV.8.2 Symplecticity and Symmetry 611
XV.8.3 Growth Parameters 614
XV.9 Exercises 615
Bibliography 617
Index 637
|
| adam_txt |
I. Examples and Numerical Experiments 1
1.1 First Problems and Methods 1
1.1.1 The Lotka Volterra Model 1
1.1.2 First Numerical Methods 3
1.1.3 The Pendulum as a Hamiltonian System 4
1.1.4 The Störmer Verlet Scheme 7
1.2 The Kepler Problem and the Outer Solar System 8
1.2.1 Angular Momentum and Kepler's Second Law 9
1.2.2 Exact Integration of the Kepler Problem 10
1.2.3 Numerical Integration of the Kepler Problem 12
1.2.4 The Outer Solar System 13
1.3 The Henon Heiles Model 15
1.4 Molecular Dynamics 18
1.5 Highly Oscillatory Problems 21
1.5.1 A Fermi Pasta Ulam Problem 21
1.5.2 Application of Classical Integrators 23
1.6 Exercises 24
II. Numerical Integrators 27
II. 1 Runge Kutta and Collocation Methods 27
II. 1.1 Runge Kutta Methods 28
II. 1.2 Collocation Methods 30
II. 1.3 Gauss and Lobatto Collocation 34
II. 1.4 Discontinuous Collocation Methods 35
11.2 Partitioned Runge Kutta Methods 38
11.2.1 Definition and First Examples 38
11.2.2 Lobatto IIIA IIIB Pairs 40
11.2.3 Nyström Methods 41
11.3 TheAdjointofaMethod 42
11.4 Composition Methods 43
11.5 Splitting Methods 47
11.6 Exercises 50
III. Order Conditions, Trees and B Series 51
III. 1 Runge Kutta Order Conditions and B Series 51
III. 1.1 Derivation of the Order Conditions 51
III. 1.2 B Series 56
III. 1.3 Composition of Methods 59
III. 1.4 Composition of B Series 61
III. 1.5 The Butcher Group 64
111.2 Order Conditions for Partitioned Runge Kutta Methods 66
111.2.1 Bi Coloured Trees and P Series 66
111.2.2 Order Conditions for Partitioned Runge Kutta Methods 68
111.2.3 Order Conditions for Nyström Methods 69
111.3 Order Conditions for Composition Methods 71
111.3.1 Introduction 71
111.3.2 The General Case 73
111.3.3 Reduction of the Order Conditions 75
111.3.4 Order Conditions for Splitting Methods 80
111.4 The Baker Campbell Hausdorff Formula 83
111.4.1 Derivative of the Exponential and Its Inverse 83
111.4.2 The BCH Formula 84
111.5 Order Conditions via the BCH Formula 87
111.5.1 Calculus of Lie Derivatives 87
111.5.2 Lie Brackets and Commutativity 89
111.5.3 Splitting Methods 91
111.5.4 Composition Methods 92
111.6 Exercises 95
IV. Conservation of First Integrals and Methods on Manifolds 97
IV. 1 Examples of First Integrals 97
IV.2 Quadratic Invariants 101
IV.2.1 Runge Kutta Methods 101
IV.2.2 Partitioned Runge Kutta Methods 102
IV.2.3 Nyström Methods 104
IV.3 Polynomial Invariants 105
IV.3.1 The Determinant as a First Integral 105
IV.3.2 Isospectral Flows 107
IV.4 Projection Methods 109
IV.5 Numerical Methods Based on Local Coordinates 113
IV.5.1 Manifolds and the Tangent Space 114
IV.5.2 Differential Equations on Manifolds 115
IV.5.3 Numerical Integrators on Manifolds 116
IV.6 Differential Equations on Lie Groups 118
IV.7 Methods Based on the Magnus Series Expansion 121
IV.8 Lie Group Methods 123
IV.8.1 Crouch Grossman Methods 124
IV.8.2 Munthe Kaas Methods 125
IV.8.3 Further Coordinate Mappings 128
IV.9 Geometrie Numerical Integration Meets Geometrie Numerical
Linear Algebra 131
IV.9.1 Numerical Integration on the Stiefel Manifold 131
IV.9.2 Differential Equations on the Grassmann Manifold . 135
IV.9.3 Dynamical Low Rank Approximation 137
IV.10 Exercises 139
V. Symmetrie Integration and Reversibility 143
VI Reversible Differential Equations and Maps 143
V.2 Symmetrie Runge Kutta Methods 146
V.2.1 Collocation and Runge Kutta Methods 146
V.2.2 Partitioned Runge Kutta Methods 148
V.3 Symmetrie Composition Methods 149
V.3.1 Symmetrie Composition of First Order Methods 150
V.3.2 Symmetrie Composition of Symmetrie Methods 154
V.3.3 Effective Order and Processing Methods 158
V.4 Symmetrie Methods on Manifolds 161
V.4.1 Symmetrie Projection 161
V.4.2 Symmetrie Methods Based on Local Coordinates 166
V.5 Energy Momentum Methods and Discrete Gradients 171
V.6 Exercises 176
VI. Symplectic Integration of Hamiltonian Systems 179
VI.l Hamiltonian Systems 180
VI. 1.1 Lagrange's Equations 180
VI. 1.2 Hamilton's Canonical Equations 181
VI.2 Symplectic Transformations 182
VI.3 First Examples of Symplectic Integrators 187
VI.4 Symplectic Runge Kutta Methods 191
VI.4.1 Criterion of Symplecticity 191
VI.4.2 Connection Between Symplectic and Symmetrie
Methods 194
VI.5 Generating Functions 195
VI.5.1 Existence of Generating Functions 195
VI.5.2 Generating Function for Symplectic Runge Kutta
Methods 198
VI.5.3 The Hamilton Jacobi Partial Differential Equation 200
VI.5.4 Methods Based on Generating Functions 203
VI.6 Variational Integrators 204
VI.6.1 Hamilton's Principle 204
VI.6.2 Discretization of Hamilton's Principle 206
VI.6.3 Symplectic Partitioned Runge Kutta Methods
Revisited 208
VI.6.4 Noether's Theorem 210
VI.7 Characterization of Symplectic Methods 212
VI.7.1 B Series Methods Conserving Quadratic First Integrals 212
VI.7.2 Characterization of Symplectic P Series (and B Series) 217
VI.7.3 Irreducible Runge Kutta Methods 220
VI.7.4 Characterization of Irreducible Symplectic Methods . 222
VI.8 Conjugate Symplecticity 222
VI.8.1 Examples and Order Conditions 223
VI.8.2 Near Conservation of Quadratic First Integrals 225
VI.9 Volume Preservation 227
VI.10 Exercises 233
VII. Non Canonical Hamiltonian Systems 237
VII. 1 Constrained Mechanical Systems 237
VII. 1.1 Introduction and Examples 237
VII. 1.2 Hamiltonian Formulation 239
VII. 1.3 A Symplectic First Order Method 242
VII. 1.4 SHAKE and RATTLE 245
VII. 1.5 The Lobatto IIIA IIIB Pair 247
VII. 1.6 Splitting Methods 252
VII.2 Poisson Systems 254
VII.2.1 Canonical Poisson Structure 254
VII.2.2 General Poisson Structures 256
VII.2.3 Hamiltonian Systems on Symplectic Submanifolds. 258
VII.3 The Darboux Lie Theorem 261
VII.3.1 Commutativity of Poisson Flows and Lie Brackets . 261
VII.3.2 Simultaneous Linear Partial Differential Equations . 262
VII.3.3 Coordinate Changes and the Darboux Lie Theorem . 265
VII.4 Poisson Integrators 268
VII.4.1 Poisson Maps and Symplectic Maps 268
VII.4.2 Poisson Integrators 270
VII.4.3 Integrators Based on the Darboux Lie Theorem 272
VII.5 Rigid Body Dynamics and Lie Poisson Systems 274
VII.5.1 History of the Euler Equations 275
VII.5.2 Hamiltonian Formulation of Rigid Body Motion 278
VII.5.3 Rigid Body Integrators 280
VII.5.4 Lie Poisson Systems 286
VII.5.5 Lie Poisson Reduction 289
VII.6 Reduced Models of Quantum Dynamics 293
VII.6.1 Hamiltonian Structure of the Schrödinger Equation . 293
VII.6.2 The Dirac Frenkel Variational Principle 295
VII.6.3 Gaussian Wavepacket Dynamics 296
VII.6.4 A Splitting Integrator for Gaussian Wavepackets 298
VII.7 Exercises 301
VIII. Structure Preserving Implementation 303
VIII. 1 Dangers of Using Standard Step Size Control 303
VIII.2 Time Transformations 306
VIII.2.1 Symplectic Integration 306
VIII.2.2 Reversible Integration 309
VIII.3 Structure Preserving Step Size Control 310
VIII.3.1 Proportional, Reversible Controllers 310
VIII.3.2 Integrating, Reversible Controllers 314
VIII.4 Multiple Time Stepping 316
VIII.4.1 Fast Slow Splitting: the Impulse Method 317
VIII.4.2 Averaged Forces 319
VIII.5 Reducing Rounding Errors 322
VIII.6 Implementation of Implicit Methods 325
VIII.6.1 Starting Approximations 326
VIII.6.2 Fixed Point Versus Newton Iteration 330
VIII.7 Exercises 335
IX. Backward Error Analysis and Structure Preservation 337
IX. 1 Modified Differential Equation Examples 337
IX.2 Modified Equations of Symmetrie Methods 342
IX. 3 Modified Equations of Symplectic Methods 343
IX.3.1 Existence of a Local Modified Hamiltonian 343
IX.3.2 Existence of a Global Modified Hamiltonian 344
IX.3.3 Poisson Integrators 347
IX.4 Modified Equations of Splitting Methods 348
IX.5 Modified Equations of Methods on Manifolds 350
IX.5.1 Methods on Manifolds and First Integrals 350
IX.5.2 Constrained Hamiltonian Systems 352
IX.5.3 Lie Poisson Integrators 354
IX.6 Modified Equations for Variable Step Sizes 356
IX.7 Rigorous Estimates Local Error 358
IX.7.1 Estimation of the Derivatives of the Numerical Solution 360
IX.7.2 Estimation of the Coefficients of the Modified Equation 362
IX.7.3 Choice of TV and the Estimation of the Local Error 364
IX.8 Long Time Energy Conservation 366
IX.9 Modified Equation in Terms of Trees 369
IX.9.1 B Series of the Modified Equation 369
IX.9.2 Elementary Hamiltonians 373
IX.9.3 Modified Hamiltonian 375
IX.9.4 First Integrals Close to the Hamiltonian 375
IX.9.5 Energy Conservation: Examples and Counter Examples 379
IX.10 Extension to Partitioned Systems 381
IX.10.1 P Series of the Modified Equation 381
IX.10.2 Elementary Hamiltonians 384
IX.ll Exercises 386
X. Hamiltonian Perturbation Theory and Symplectic Integrators 389
X. 1 Completely Integrable Hamiltonian Systems 390
X. 1.1 Local Integration by Quadrature 390
X.1.2 Completely Integrable Systems 393
X.1.3 Action Angle Variables 397
X.1.4 Conditionally Periodic Flows 399
X. 1.5 The Toda Lattice an Integrable System 402
X.2 Transformations in the Perturbation Theory for Integrable
Systems 404
X.2.1 The Basic Scheme of Classical Perturbation Theory . 405
X.2.2 Lindstedt Poincare Series 406
X.2.3 Kolmogorov's Iteration 410
X.2.4 Birkhoff Normalization Near an Invariant Torus 412
X.3 Linear Error Growth and Near Preservation of First Integrals . 413
X.4 Near Invariant Tori on Exponentially Long Times 417
X.4.1 Estimates of Perturbation Series 417
X.4.2 Near Invariant Tori of Perturbed Integrable Systems . 421
X.4.3 Near Invariant Tori of Symplectic Integrators 422
X.5 Kolmogorov's Theorem on Invariant Tori 423
X.5.1 Kolmogorov's Theorem 423
X.5.2 KAM Tori under Symplectic Discretization 428
X.6 Invariant Tori of Symplectic Maps 430
X.6.1 A KAM Theorem for Symplectic Near Identity Maps . 431
X.6.2 Invariant Tori of Symplectic Integrators 433
X.6.3 Strongly Non Resonant Step Sizes 433
X.7 Exercises 434
XI. Reversible Perturbation Theory and Symmetrie Integrators 437
XI.l Integrable Reversible Systems 437
XI.2 Transformations in Reversible Perturbation Theory 442
XI.2.1 The Basic Scheme of Reversible Perturbation Theory. 443
XI.2.2 Reversible Perturbation Series 444
XI.2.3 Reversible KAM Theory 445
XI.2.4 Reversible Birkhoff Type Normalization 447
XI.3 Linear Error Growth and Near Preservation of First Integrals . 448
XI.4 Invariant Tori under Reversible Discretization 451
XI.4.1 Near Invariant Tori over Exponentially Long Times . 451
XI.4.2 A KAM Theorem for Reversible Near Identity Maps . 451
XI.5 Exercises 453
XII. Dissipatively Perturbed Hamiltonian and Reversible Systems 455
XII. 1 Numerical Experiments with Van der Pol's Equation 455
XII.2 Averaging Transformations 458
XII.2.1 The Basic Scheme of Averaging 458
XII.2.2 Perturbation Series 459
XII.3 Attractive Invariant Manifolds 460
XII.4 Weakly Attractive Invariant Tori of Perturbed Integrable Systems 464
XII.5 Weakly Attractive Invariant Tori of Numerical Integrators 465
XII.5.1 Modified Equations of Perturbed Differential Equations 466
XII.5.2 Symplectic Methods 467
XII.5.3 Symmetrie Methods 469
XII.6 Exercises 469
XIII. Oscillatory Differential Equations with Constant High Frequencies. 471
XIII. 1 Towards Longer Time Steps in Solving Oscillatory Equations
of Motion 471
XIII. 1.1 The Stornier Verlet Method vs. Multiple Time Scales .472
XIII. 1.2 Gautschi's and Deuflhard's Trigonometrie Methods . 473
XIII.1.3 The Impulse Method 475
XIII.1.4 The Mollified Impulse Method 476
XIII. 1.5 Gautschi's Method Revisited 477
XIII.1.6 Two Force Methods 478
XIII.2 A Nonlinear Model Problem and Numerical Phenomena 478
XIII.2.1 Time Scales in the Fermi Pasta Ulam Problem 479
XIII.2.2 Numerical Methods 481
XIII.2.3 Accuracy Comparisons 482
XIII.2.4 Energy Exchange between Stiff Components 483
XIII.2.5 Near Conservation of Total and Oscillatory Energy. 484
XIII.3 Principal Terms of the Modulated Fourier Expansion 486
XIII.3.1 Decomposition of the Exact Solution 486
XIII.3.2 Decomposition of the Numerical Solution 488
XIII.4 Accuracy and Slow Exchange 490
XIII.4.1 Convergence Properties on Bounded Time Intervals . 490
XIII.4.2 Intra Oscillatory and Oscillatory Smooth Exchanges . 494
XIII.5 Modulated Fourier Expansions 496
XIII.5.1 Expansion of the Exact Solution 496
XIII.5.2 Expansion of the Numerical Solution 498
XIII.5.3 Expansion of the Velocity Approximation 502
XIII.6 Almost Invariants of the Modulated Fourier Expansions 503
XIII.6.1 The Hamiltonian of the Modulated Fourier Expansion . 503
XIII.6.2 A Formal Invariant Close to the Oscillatory Energy . 505
XIII.6.3 Almost Invariants of the Numerical Method 507
XIII.7 Long Time Near Conservation of Total and Oscillatory Energy .510
XIII.8 Energy Behaviour of the Störmer Verlet Method 513
XIII.9 Systems with Several Constant Frequencies 516
XIII.9.1 Oscillatory Energies and Resonances 517
XIII.9.2 Multi Frequency Modulated Fourier Expansions 519
XIII.9.3 Almost Invariants of the Modulation System 521
XIII.9.4 Long Time Near Conservation of Total and
Oscillatory Energies 524
XIII. 10 Systems with Non Constant Mass Matrix 526
XIII.ll Exercises 529
XIV. Oscülatory Differential Equations with Varying High Frequencies .531
XIV. 1 Linear Systems with Time Dependent Skew Hermitian Matrix .531
XIV. 1.1 Adiabatic Transformation and Adiabatic Invariants . 531
XIV.1.2 Adiabatic Integrators 536
XIV.2 Mechanical Systems with Time Dependent Frequencies 539
XIV.2.1 Canonical Transformation to Adiabatic Variables 540
XIV.2.2 Adiabatic Integrators 547
XIV.2.3 Error Analysis of the Impulse Method 550
XIV.2.4 Error Analysis of the Mollified Impulse Method 554
XIV.3 Mechanical Systems with Solution Dependent Frequencies 555
XIV.3.1 Constraining Potentials 555
XIV.3.2 Transformation to Adiabatic Variables 558
XIV.3.3 Integrators in Adiabatic Variables 563
XIV.3.4 Analysis of Multiple Time Stepping Methods 564
XIV.4 Exercises 564
XV. Dynamics of Multistep Methods 567
XV. 1 Numerical Methods and Experiments 567
XV.1.1 Linear Multistep Methods 567
XV. 1.2 Multistep Methods for Second Order Equations 569
XV. 1.3 Partitioned Multistep Methods 572
XV.2 The Underlying One Step Method 573
XV.2.1 Strictly Stable Multistep methods 573
XV.2.2 Formal Analysis for Weakly Stable Methods 575
XV.3 Backward Error Analysis 576
XV.3.1 Modified Equation for Smooth Numerical Solutions . 576
XV.3.2 Parasitic Modified Equations 579
XV.4 Can Multistep Methods be Symplectic? 585
XV.4.1 Non Symplecticity of the Underlying One Step Method 585
XV.4.2 Symplecticity in the Higher Dimensional Phase Space . 587
XV.4.3 Modified Hamiltonian of Multistep Methods 589
XV.4.4 Modified Quadratic First Integrals 591
XV.5 Long Term Stability 592
XV.5.1 Role of Growth Parameters 592
XV.5.2 Hamiltonian of the Füll Modified System 594
XV.5.3 Long Time Bounds for Parasitic Solution Components 596
XV.6 Explanation of the Long Time Behaviour 600
XV.6.1 Conservation of Energy and Angular Momentum 600
XV.6.2 Linear Error Growth for Integrable Systems 601
XV.7 Practical Considerations 602
XV.7.1 Numerical Instabilities and Resonances 602
XV.7.2 Extension to Variable Step Sizes 605
XV. 8 Multi Value or General Linear Methods 609
XV.8.1 Underlying One Step Method and Backward Error
Analysis 609
XV.8.2 Symplecticity and Symmetry 611
XV.8.3 Growth Parameters 614
XV.9 Exercises 615
Bibliography 617
Index 637 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Hairer, Ernst 1949- Lubich, Christian 1959- Wanner, Gerhard 1942- |
| author_GND | (DE-588)139445188 (DE-588)11167090X (DE-588)13944534X |
| author_facet | Hairer, Ernst 1949- Lubich, Christian 1959- Wanner, Gerhard 1942- |
| author_role | aut aut aut |
| author_sort | Hairer, Ernst 1949- |
| author_variant | e h eh c l cl g w gw |
| building | Verbundindex |
| bvnumber | BV022303806 |
| classification_rvk | SK 920 |
| classification_tum | MAT 000 MAT 665f |
| collection | ZDB-2-SMA |
| ctrlnum | (OCoLC)873394332 (DE-599)BVBBV022303806 |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| doi_str_mv | 10.1007/3-540-30666-8 |
| edition | 2. ed. |
| format | Electronic eBook |
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| id | DE-604.BV022303806 |
| illustrated | Illustrated |
| index_date | 2024-07-02T16:55:57Z |
| indexdate | 2024-07-09T20:54:34Z |
| institution | BVB |
| isbn | 9783540306665 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-015513742 |
| oclc_num | 873394332 |
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| owner_facet | DE-739 DE-355 DE-BY-UBR DE-634 DE-91 DE-BY-TUM DE-384 DE-703 DE-83 DE-19 DE-BY-UBM |
| physical | 1 Online-Ressource (XVII, 644 S.) Ill., graph. Darst. |
| psigel | ZDB-2-SMA |
| publishDate | 2006 |
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| publisher | Springer |
| record_format | marc |
| series | Springer series in computational mathematics |
| series2 | Springer series in computational mathematics |
| spelling | Hairer, Ernst 1949- Verfasser (DE-588)139445188 aut Geometric numerical integration structure-preserving algorithms for ordinary differential equations Ernst Hairer ; Christian Lubich ; Gerhard Wanner 2. ed. Berlin [u.a.] Springer 2006 1 Online-Ressource (XVII, 644 S.) Ill., graph. Darst. txt rdacontent c rdamedia cr rdacarrier Springer series in computational mathematics 31 Numerische Integration (DE-588)4172168-8 gnd rswk-swf Gewöhnliche Differentialgleichung (DE-588)4020929-5 gnd rswk-swf Gewöhnliche Differentialgleichung (DE-588)4020929-5 s Numerische Integration (DE-588)4172168-8 s DE-604 Lubich, Christian 1959- Verfasser (DE-588)11167090X aut Wanner, Gerhard 1942- Verfasser (DE-588)13944534X aut Erscheint auch als Druckausgabe 3-540-30663-3 Erscheint auch als Druckausgabe 978-3-540-30663-4 Springer series in computational mathematics 31 (DE-604)BV035421315 31 https://doi.org/10.1007/3-540-30666-8 Verlag Volltext HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015513742&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Hairer, Ernst 1949- Lubich, Christian 1959- Wanner, Gerhard 1942- Geometric numerical integration structure-preserving algorithms for ordinary differential equations Springer series in computational mathematics Numerische Integration (DE-588)4172168-8 gnd Gewöhnliche Differentialgleichung (DE-588)4020929-5 gnd |
| subject_GND | (DE-588)4172168-8 (DE-588)4020929-5 |
| title | Geometric numerical integration structure-preserving algorithms for ordinary differential equations |
| title_auth | Geometric numerical integration structure-preserving algorithms for ordinary differential equations |
| title_exact_search | Geometric numerical integration structure-preserving algorithms for ordinary differential equations |
| title_exact_search_txtP | Geometric numerical integration structure-preserving algorithms for ordinary differential equations |
| title_full | Geometric numerical integration structure-preserving algorithms for ordinary differential equations Ernst Hairer ; Christian Lubich ; Gerhard Wanner |
| title_fullStr | Geometric numerical integration structure-preserving algorithms for ordinary differential equations Ernst Hairer ; Christian Lubich ; Gerhard Wanner |
| title_full_unstemmed | Geometric numerical integration structure-preserving algorithms for ordinary differential equations Ernst Hairer ; Christian Lubich ; Gerhard Wanner |
| title_short | Geometric numerical integration |
| title_sort | geometric numerical integration structure preserving algorithms for ordinary differential equations |
| title_sub | structure-preserving algorithms for ordinary differential equations |
| topic | Numerische Integration (DE-588)4172168-8 gnd Gewöhnliche Differentialgleichung (DE-588)4020929-5 gnd |
| topic_facet | Numerische Integration Gewöhnliche Differentialgleichung |
| url | https://doi.org/10.1007/3-540-30666-8 http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015513742&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV035421315 |
| work_keys_str_mv | AT hairerernst geometricnumericalintegrationstructurepreservingalgorithmsforordinarydifferentialequations AT lubichchristian geometricnumericalintegrationstructurepreservingalgorithmsforordinarydifferentialequations AT wannergerhard geometricnumericalintegrationstructurepreservingalgorithmsforordinarydifferentialequations |