Geometric numerical integration: structure-preserving algorithms for ordinary differential equations
Gespeichert in:
| Beteiligte Personen: | , , |
|---|---|
| Format: | Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
Berlin [u.a.]
Springer
2006
|
| Ausgabe: | 2. ed. |
| Schriftenreihe: | Springer series in computational mathematics
31 |
| Schlagwörter: | |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014742360&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Abstract: | 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. |
| Umfang: | XVII, 644 S. Ill., graph. Darst. |
| ISBN: | 3540306633 9783540306634 |
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. | ||
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2006 | |
| 300 | |a XVII, 644 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Springer series in computational mathematics |v 31 | |
| 520 | 3 | |a 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. | |
| 650 | 7 | |a Análise numérica |2 larpcal | |
| 650 | 7 | |a Equações diferenciais |2 larpcal | |
| 650 | 7 | |a Integração |2 larpcal | |
| 650 | 7 | |a Sistemas hamiltonianos |2 larpcal | |
| 650 | 4 | |a Differential equations |x Numerical solutions | |
| 650 | 4 | |a Hamiltonian systems | |
| 650 | 4 | |a Numerical integration | |
| 650 | 0 | 7 | |a Gewöhnliche Differentialgleichung |0 (DE-588)4020929-5 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Numerische Integration |0 (DE-588)4172168-8 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Gewöhnliche Differentialgleichung |0 (DE-588)4020929-5 |D s |
| 689 | 0 | 1 | |a Numerische Integration |0 (DE-588)4172168-8 |D s |
| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Lubich, Christian |d 1959- |e Verfasser |0 (DE-588)11167090X |4 aut | |
| 700 | 1 | |a Wanner, Gerhard |e Verfasser |4 aut | |
| 830 | 0 | |a Springer series in computational mathematics |v 31 |w (DE-604)BV000012004 |9 31 | |
| 856 | 4 | 2 | |m Digitalisierung UB Augsburg |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014742360&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 943 | 1 | |a oai:aleph.bib-bvb.de:BVB01-014742360 | |
Datensatz im Suchindex
| DE-BY-TUM_call_number | 0102 MAT 665 2006 A 4255(2) 0102 MAT 665f 2006 A 4255(2) |
|---|---|
| DE-BY-TUM_katkey | 1544744 |
| DE-BY-TUM_location | 01 |
| DE-BY-TUM_media_number | 040020294435 040010087875 |
| _version_ | 1821933055971950592 |
| adam_text | Table
of
Contents
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
Stornier-
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
Hénon-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
1IIA-IIIB
Pairs
........................... 40
H.2.3
Nyström
Methods
............................... 41
11.3 The Adjoint of a Method
................................. 42
11.4 Composition Methods
.................................... 43
11.
5
Splitting Methods
....................................... 47
11.6 Exercises
............................................... 50
χ
Table of Contents
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
III.
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
Ш.З
Order Conditions for Composition Methods
................. 71
Ш.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
1II.4 The Baker-Campbell-Hausdorff Formula
.................... 83
Ш.4.
1
Derivative of the Exponential and Its Inverse
......... 83
III.4.2 The BCH Formula
............................... 84
ΠΙ.5
Order Conditions via the BCH Formula
..................... 87
ΙΪΙ.5.
1
Calculus of Lie Derivatives
........................ 87
1II.5.2 Lie Brackets and Commutativity
................... 89
ITI.5.3 Splitting Methods
................................ 91
III.5.4 Composition Methods
............................ 92
III.6 Exercises
............................................... 95
IV. Conservation of First Integrals and Methods on Manifolds
........ 97
IV.
1
Examples of First Integrals
................................ 97
I V.2 Quadratic Invariants
.....................................101
IV.2.
1
Runge-Kutta Methods
............................101
IV.2.2 Partitioned Runge-Kutta Methods
..................102
IV.2.3
Nyström
Methods
...............................
1
04
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
I V.5 Numerical Methods Based on Local Coordinates
.............113
IV.5.1 Manifolds and the Tangent Space
...................114
1V.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
1V.8.
1
Crouch-Grossman Methods
.......................124
IV.8.2 Munthe-Kaas Methods
...........................125
Table
of Contents
xi
IV.8.3 Further Coordinate Mappings
......................128
IV.9 Geometric Numerical Integration Meets Geometric Numerical
Linear Algebra
..........................................131
IV.9.
1
Numerical Integration on the
Stiefel
Manifold
........131
1V.9.2 Differential Equations on the
Grassmann
Manifold
.... 135
IV.9.3 Dynamical Low-Rank Approximation
...............137
1V.10 Exercises
...............................................139
V. Symmetric Integration and Reversibility
....................... 143
V.I Reversible Differential Equations and Maps
.................143
V.2 Symmetric Runge-Kutta Methods
..........................146
V.2.
1
Collocation and Runge-Kutta Methods
..............146
V.2.2 Partitioned Runge-Kutta Methods
..................148
V.3 Symmetric Composition Methods
..........................149
V.3.
1
Symmetric Composition of First Order Methods
......150
V.3.
2
Symmetric Composition of Symmetric Methods
......154
V.3.3 Effective Order and Processing Methods
............158
V.4 Symmetric Methods on Manifolds
..........................161
V.4.
1
Symmetric Projection
............................161
V.4.
2
Symmetric 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.
1
Hamiltonian Systems
....................................180
■
VI.
1.1
Lagrange s Equations
............................180
VI.
1.2
Hamilton s Canonical Equations
...................181
V1.2 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 Symmetric
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
xii
Table
of Contents
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 I1IA
-
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
V1I.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
TabJe of Contents
xiii
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
VII1.4.1 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 Symmetric 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
1X.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
N
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
1X.1
0.1
P-Series of the Modified Equation
..................381
IX.
10.2
Elementary Hamiltonians
.........................384
IX.
11
Exercises
...............................................386
xiv
Table
of
Contents
X.
Hamiltonian Perturbation
Theory and Symplectic Integrators
.....389
Χ.
1
Completely
Integrable Hamiltonian Systems
.................390
X.
1.1
Local
Integration by Quadrature
...................390
Χ.
1.2
Completely
Integrable Systems
....................393
•
X.1.3
Action-Angle
Variables
...........................397
X.1.4
Conditionally Periodic Flows
......................399
X.I
.5
The
Toda
Lattice
-
an
Integrable
System
............402
X.2 Transformations in the Perturbation Theory for
Integrable
Systems
................................................404
X.2.
і
The Basic Scheme of Classical Perturbation Theory
. .. 405
X.2.
2
Lindstedt-Poincaré
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
XL Reversible Perturbation Theory and Symmetric Integrators
.......437
XI.
1
Integrable
Reversible Systems
.............................437
XI.
2
Transformations in Reversible Perturbation Theory
...........442
X1.2.
1
The Basic Scheme of Reversible Perturbation Theory
.. 443
XI.2.2 Reversible Perturbation Series
.....................444
XI.2.3 Reversible
KAM
Theory
..........................445
Xl.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
Xl.4.2
A
KAM
Theorem for Reversible Near-Identity Maps
.. 451
X1.5 Exercises
...............................................453
XII.
Dissipatively Perturbed Hamiltonian and Reversible Systems
......455
XII.l 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
Table
of
Contents xv
XII.
3
Attractive Invariant Manifolds
.............................460
XII.4
Weakly Attractive Invariant Tori of Perturbed
Integrable
Systems
464
ХП.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 Symmetric 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
Deuflharďs
Trigonometric 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
ХШ.2.2
Numerical Methods
..............................481
XIII.2.3
Accuracy Comparisons
...........................482
ХШ.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
XI11.5 Modulated Fourier Expansions
............................496
XIII.5.1
Expansion of the Exact Solution
...................496
ХПІ.5.2
Expansion of the Numerical Solution
...............498
XIII.5.3
Expansion of the Velocity Approximation
...........502
ХШ.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
Stornier-
Verlet
Method
.............513
XIII.9
Systems with Several Constant Frequencies
..................516
ХШ.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
xvi
Table
of Contents
XIII.
10
Systems with Non-Constant Mass Matrix
....................526
XIII. 11
Exercises
...............................................529
XIV.
Oscillatory Differential Equations with Varying High Frequencies
. . 531
XIV.l 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.
!
Role of Growth Parameters
........................592
XV.5.2 Hamiltonian of the Full 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
Table
of Contents
xvii
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 |
| author | Hairer, Ernst 1949- Lubich, Christian 1959- Wanner, Gerhard |
| author_GND | (DE-588)139445188 (DE-588)11167090X |
| author_facet | Hairer, Ernst 1949- Lubich, Christian 1959- Wanner, Gerhard |
| author_role | aut aut aut |
| author_sort | Hairer, Ernst 1949- |
| author_variant | e h eh c l cl g w gw |
| building | Verbundindex |
| bvnumber | BV021525932 |
| callnumber-first | Q - Science |
| callnumber-label | QA299 |
| callnumber-raw | QA299.3 |
| callnumber-search | QA299.3 |
| callnumber-sort | QA 3299.3 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 920 |
| classification_tum | MAT 665f |
| ctrlnum | (OCoLC)69223213 (DE-599)BVBBV021525932 |
| dewey-full | 515/.352 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515/.352 |
| dewey-search | 515/.352 |
| dewey-sort | 3515 3352 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 2. ed. |
| format | Book |
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| id | DE-604.BV021525932 |
| illustrated | Illustrated |
| indexdate | 2024-12-20T12:35:40Z |
| institution | BVB |
| isbn | 3540306633 9783540306634 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-014742360 |
| oclc_num | 69223213 |
| open_access_boolean | |
| owner | DE-91G DE-BY-TUM DE-824 DE-29T DE-384 DE-706 DE-83 DE-11 DE-634 DE-188 |
| owner_facet | DE-91G DE-BY-TUM DE-824 DE-29T DE-384 DE-706 DE-83 DE-11 DE-634 DE-188 |
| physical | XVII, 644 S. Ill., graph. Darst. |
| publishDate | 2006 |
| publishDateSearch | 2006 |
| publishDateSort | 2006 |
| publisher | Springer |
| record_format | marc |
| series | Springer series in computational mathematics |
| series2 | Springer series in computational mathematics |
| spellingShingle | Hairer, Ernst 1949- Lubich, Christian 1959- Wanner, Gerhard Geometric numerical integration structure-preserving algorithms for ordinary differential equations Springer series in computational mathematics Análise numérica larpcal Equações diferenciais larpcal Integração larpcal Sistemas hamiltonianos larpcal Differential equations Numerical solutions Hamiltonian systems Numerical integration Gewöhnliche Differentialgleichung (DE-588)4020929-5 gnd Numerische Integration (DE-588)4172168-8 gnd |
| subject_GND | (DE-588)4020929-5 (DE-588)4172168-8 |
| 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_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 | Análise numérica larpcal Equações diferenciais larpcal Integração larpcal Sistemas hamiltonianos larpcal Differential equations Numerical solutions Hamiltonian systems Numerical integration Gewöhnliche Differentialgleichung (DE-588)4020929-5 gnd Numerische Integration (DE-588)4172168-8 gnd |
| topic_facet | Análise numérica Equações diferenciais Integração Sistemas hamiltonianos Differential equations Numerical solutions Hamiltonian systems Numerical integration Gewöhnliche Differentialgleichung Numerische Integration |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014742360&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000012004 |
| work_keys_str_mv | AT hairerernst geometricnumericalintegrationstructurepreservingalgorithmsforordinarydifferentialequations AT lubichchristian geometricnumericalintegrationstructurepreservingalgorithmsforordinarydifferentialequations AT wannergerhard geometricnumericalintegrationstructurepreservingalgorithmsforordinarydifferentialequations |
Inhaltsverzeichnis
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Teilbibliothek Mathematik & Informatik
| Signatur: |
0102 MAT 665 2006 A 4255(2) Lageplan 0102 MAT 665f 2006 A 4255(2) Lageplan |
|---|---|
| Exemplar 1 | Ausleihbar Am Standort |
| Exemplar 2 | Ausleihbar Am Standort |