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Geometric Numerical Integration: Structure-preserving Algorithms for Ordinary Differential Equations [Kõva köide]

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  • Formaat: Hardback, 528 pages, kõrgus x laius x paksus: 230x159x25 mm, kaal: 885 g, illustrations, bibliog
  • Sari: Springer Series in Computational Mathematics v.31
  • Ilmumisaeg: 01-May-2002
  • Kirjastus: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • ISBN-10: 3540430032
  • ISBN-13: 9783540430032
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Geometric Numerical Integration: Structure-preserving Algorithms for Ordinary Differential EquationsSuurem pilt
  • Formaat: Hardback, 528 pages, kõrgus x laius x paksus: 230x159x25 mm, kaal: 885 g, illustrations, bibliog
  • Sari: Springer Series in Computational Mathematics v.31
  • Ilmumisaeg: 01-May-2002
  • Kirjastus: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • ISBN-10: 3540430032
  • ISBN-13: 9783540430032
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9783540430032.html
Märksõnad:
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.
Examples and Numerical Experiments
1(22)
Two - Dimensional Problems
1(6)
The Lotka-Volterra Model
1(3)
Hamiltonian Systems - the Pendulum
4(3)
The Kepler Problem and the Outer Solar System
7(5)
Exact Integration of the Kepler Problem
7(2)
Numerical Integration of the Kepler Problem
9(1)
The Outer Solar System
10(2)
Molecular Dynamics
12(4)
The Stormer/Verlet Scheme
13(2)
Numerical Experiments
15(1)
Highly Oscillatory Problems
16(4)
A Fermi-Pasta-Ulam Problem
17(1)
Application of Classical Integrators
18(2)
Exercises
20(3)
Numerical Integrators
23(24)
Runge-Kutta and Collocation Methods
23(11)
Runge-Kutta Methods
24(2)
Collocation Methods
26(4)
Gauss and Lobatto Collocation
30(1)
Discontinuous Collocation Methods
31(3)
Partitioned Runge-Kutta Methods
34(4)
Definition and First Examples
34(2)
Lobatto IIIA - IIIB Pairs
36(1)
Nystrom Methods
37(1)
The Adjoint of a Method
38(1)
Composition Methods
39(2)
Splitting Methods
41(5)
Exercises
46(1)
Order Conditions, Trees and B-Series
47(46)
Runge-Kutta Order Conditions and B-Series
47(15)
Derivation of the Order Conditions
47(5)
B-Series
52(3)
Composition of Methods
55(2)
Composition of B-Series
57(3)
The Butcher Group
60(2)
Order Conditions for Partitioned Runge-Kutta Methods
62(5)
Bi-Coloured Trees and P-Series
62(2)
Order Conditions for Partitioned Runge-Kutta Methods
64(1)
Order Conditions for Nystrom Methods
65(2)
Order Conditions for Composition Methods
67(11)
Introduction
67(2)
The General Case
69(2)
Reduction of the Order Conditions
71(5)
Order Conditions for Splitting Methods
76(2)
The Baker-Campbell-Hausdorff Formula
78(5)
Derivative of the Exponential and Its Inverse
78(2)
The BCH Formula
80(3)
Order Conditions via the BCH Formula
83(7)
Calculus of Lie Derivatives
83(2)
Lie Brackets and Commutativity
85(1)
Splitting Methods
86(2)
Composition Methods
88(2)
Exercises
90(3)
Conservation of First Integrals and Methods on Manifolds
93(38)
Examples of First Integrals
93(4)
Quadratic Invariants
97(4)
Runge-Kutta Methods
97(1)
Partitioned Runge-Kutta Methods
98(2)
Nystrom Methods
100(1)
Polynomial Invariants
101(4)
The Determinant as a First Integral
101(2)
Isospectral Flows
103(2)
Projection Methods
105(5)
Numerical Methods Based on Local Coordinates
110(5)
Manifolds and the Tangent Space
110(2)
Differential Equations on Manifolds
112(1)
Numerical Integrators on Manifolds
112(3)
Differential Equations on Lie Groups
115(3)
Methods Based on the Magnus Series Expansion
118(3)
Lie Group Methods
121(7)
Crouch-Grossman Methods
121(2)
Munthe-Kaas Methods
123(2)
Further Coordinate Mappings
125(3)
Exercises
128(3)
Symmetric Integration and Reversibility
131(36)
Reversible Differential Equations and Maps
131(3)
Symmetric Runge-Kutta Methods
134(3)
Collocation and Runge-Kutta Methods
134(2)
Partitioned Runge-Kutta Methods
136(1)
Symmetric Composition Methods
137(12)
Symmetric Composition of First Order Methods
138(4)
Symmetric Composition of Symmetric Methods
142(4)
Effective Order and Processing Methods
146(3)
Symmetric Methods on Manifolds
149(10)
Symmetric Projection
149(5)
Symmetric Methods Based on Local Coordinates
154(5)
Energy - Momentum Methods and Discrete Gradients
159(5)
Exercises
164(3)
Symplectic Integration of Hamiltonian Systems
167(42)
Hamiltonian Systems
168(2)
Lagrange's Equations
168(1)
Hamilton's Canonical Equations
169(1)
Symplectic Transformations
170(5)
First Examples of Symplectic Integrators
175(3)
Symplectic Runge-Kutta Methods
178(4)
Criterion of Symplecticity
178(3)
Connection Between Symplectic and Symmetric Methods
181(1)
Generating Functions
182(9)
Existence of Generating Functions
182(2)
Generating Function for Symplectic Runge-Kutta Methods
184(2)
The Hamilton-Jacobi Partial Differential Equation
186(3)
Methods Based on Generating Functions
189(2)
Variational Integrators
191(8)
Hamilton's Principle
191(1)
Discretization of Hamilton's Principle
192(3)
Symplectic Partitioned Runge-Kutta Methods Revisited
195(2)
Noether's Theorem
197(2)
Characterization of Symplectic Methods
199(7)
Symplectic P-Series (and B-Series)
199(3)
Irreducible Runge-Kutta Methods
202(1)
Characterization of Irreducible Symplectic Methods
203(1)
Conjugate Symplecticity
204(2)
Exercises
206(3)
Further Topics in Structure Preservation
209(46)
Constrained Mechanical Systems
209(17)
Introduction and Examples
209(2)
Hamiltonian Formulation
211(2)
A Symplectic First Order Method
213(3)
SHAKE and RATTLE
216(2)
The Lobatto IIIA - IIIB Pair
218(6)
Splitting Methods
224(2)
Poisson Systems
226(22)
Canonical Poisson Structure
226(2)
General Poisson Structures
228(3)
Simultaneous Linear Partial Differential Equations
231(3)
Coordinate Changes and the Darboux-Lie Theorem
234(3)
Poisson Integrators
237(5)
Lie-Poisson Systems
242(6)
Volume Preservation
248(5)
Exercises
253(2)
Structure-Preserving Implementation
255(32)
Dangers of Using Standard Step Size Control
255(3)
Reversible Adaptive Step Size Selection
258(3)
Time Transformations
261(5)
Symplectic Integration
261(3)
Reversible Integration
264(2)
Multiple Time Stepping
266(6)
Fast-Slow Splitting: the Impulse Method
266(3)
Averaged Forces
269(3)
Reducing Rounding Errors
272(3)
Implementation of Implicit Methods
275(9)
Starting Approximations
275(4)
Fixed-Point Versus Newton Iteration
279(5)
Exercises
284(3)
Backward Error Analysis and Structure Preservation
287(40)
Modified Differential Equation - Examples
287(5)
Modified Equations of Symmetric Methods
292(1)
Modified Equations of Symplectic Methods
293(5)
Existence of a Local Modified Hamiltonian
293(1)
Existence of a Global Modified Hamiltonian
294(3)
Poisson Integrators
297(1)
Modified Equations of Splitting Methods
298(2)
Modified Equations of Methods on Manifolds
300(3)
Modified Equations for Variable Step Sizes
303(1)
Rigorous Estimates - Local Error
304(8)
Estimation of the Derivatives of the Numerical Solution
306(1)
Estimation of the Coefficients of the Modified Equation
307(3)
Choice of N and the Estimation of the Local Error
310(2)
Long-Time Energy Conservation
312(2)
Modified Equation in Terms of Trees
314(5)
B-Series of the Modified Equation
315(2)
Extension to Partitioned Systems
317(2)
Modified Hamiltonian
319(6)
Elementary Hamiltonians
321(3)
Characterization of Symplectic P-Series
324(1)
Exercises
325(2)
Hamiltonian Perturbation Theory and Symplectic Integrators
327(48)
Completely Integrable Hamiltonian Systems
328(14)
Local Integration by Quadrature
328(3)
Completely Integrable Systems
331(4)
Action-Angle Variables
335(2)
Conditionally Periodic Flows
337(3)
The Toda Lattice - an Integrable System
340(2)
Transformations in the Perturbation Theory for Integrable Systems
342(9)
The Basic Scheme of Classical Perturbation Theory
343(1)
Lindstedt-Poincare Series
344(4)
Kolmogorov's Iteration
348(2)
Birkhoff Normalization Near an Invariant Torus
350(1)
Linear Error Growth and Near-Preservation of First Integrals
351(4)
Near-Invariant Tori on Exponentially Long Times
355(6)
Estimates of Perturbation Series
355(4)
Near-Invariant Tori of Perturbed Integrable Systems
359(1)
Near-Invariant Tori of Symplectic Integrators
360(1)
Kolmogorov's Theorem on Invariant Tori
361(7)
Kolmogorov's Theorem
361(5)
KAM Tori under Symplectic Discretization
366(2)
Invariant Tori of Symplectic Maps
368(4)
A KAM Theorem for Symplectic Near-Identity Maps
369(2)
Invariant Tori of Symplectic Integrators
371(1)
Strongly Non-Resonant Step Sizes
371(1)
Exercises
372(3)
Reversible Perturbation Theory and Symmetric Integrators
375(16)
Integrable Reversible Systems
375(4)
Transformations in Reversible Perturbation Theory
379(5)
The Basic Scheme of Reversible Perturbation Theory
379(1)
Reversible Perturbation Series
380(2)
Reversible KAM Theory
382(2)
Reversible Birkhoff Type Normalization
384(1)
Linear Error Growth and Near-Preservation of First Integrals
384(2)
Invariant Tori under Reversible Discretization
386(3)
Near-Invariant Tori over Exponentially Long Times
386(1)
A KAM Theorem for Reversible Near-Identity Maps
387(2)
Exercises
389(2)
Dissipatively Perturbed Hamiltonian and Reversible Systems
391(16)
Numerical Experiments with Van der Pol's Equation
391(3)
Averaging Transformations
394(2)
The Basic Scheme of Averaging
394(1)
Perturbation Series
395(1)
Attractive Invariant Manifolds
396(4)
Weakly Attractive Invariant Tori of Perturbed Integrable Systems
400(1)
Weakly Attractive Invariant Tori of Numerical Integrators
401(4)
Modified Equations of Perturbed Differential Equations
402(1)
Symplectic Methods
403(2)
Symmetric Methods
405(1)
Exercises
405(2)
Highly Oscillatory Differential Equations
407(48)
Towards longer Time Steps in Solving Oscillatory Differential Equations
407(7)
The Stormer/Verlet Method vs. Multiple Time Scales
408(1)
Gautschi's and Deuflhard's Trigonometric Methods
409(2)
The Impulse Method
411(1)
The Mollified Impulse Method
412(1)
Gautschi's Method Revisited
413(1)
Two-Force Methods
414(1)
A Nonlinear Model Problem and Numerical Phenomena
414(8)
Time Scales in the Fermi-Pasta-Ulam Problem
415(1)
Numerical Methods
416(2)
Accuracy Comparisons
418(1)
Energy Exchange between Stiff Components
419(1)
Near-Conservation of Total and Oscillatory Energy
420(2)
Principal Terms of the Modulated Fourier Expansion
422(4)
Decomposition of the Exact Solution
422(2)
Decomposition of the Numerical Solution
424(2)
Accuracy and Slow Exchange
426(6)
Convergence Properties on Bounded Time Intervals
426(5)
Intra-Oscillatory and Oscillatory-Smooth Exchanges
431(1)
Modulated Fourier Expansions
432(7)
Expansion of the Exact Solution
433(2)
Expansion of the Numerical Solution
435(3)
Expansion of the Velocity Approximation
438(1)
Almost-Invariants of the Modulated Fourier Expansions
439(8)
The Hamiltonian of the Modulated Fourier Expansion
440(1)
Formal Invariant Close to the Oscillatory Energy
441(2)
Almost-Invariants of the Numerical Method
443(4)
Long-Time Near-Conservation of Total and Oscillatory Energy
447(2)
Energy Behaviour of the Stormer/Verlet Method
449(3)
Exercises
452(3)
Dynamics of Multistep Methods
455(38)
Numerical Methods and Experiments
455(6)
Linear Multistep Methods
455(2)
Multistep Methods for Second Order Equations
457(2)
Partitioned Multistep Methods
459(1)
Multi-Value or General Linear Methods
460(1)
Related One-Step Methods
461(9)
The Underlying One-Step Method
461(2)
Formal Analysis for Weakly Stable Methods
463(1)
Backward Error Analysis for Multistep Methods
464(3)
Dynamics of Weakly Stable Methods
467(1)
Invariant Manifold of the Augmented System
468(2)
Can Multistep Methods be Symplectic?
470(4)
Non-Symplecticity of the Underlying One-Step Method
470(1)
Symplecticity in the Higher-Dimensional Phase Space
471(3)
Symmetric Multi-Value Methods
474(3)
Definition of Symmetry
474(2)
A Useful Criterion for Symmetry
476(1)
Stability of the Invariant Manifold
477(13)
Partitioned General Linear Methods
477(2)
The Linearized Augmented System
479(2)
Dissipatively Perturbed Hamiltonian Systems
481(2)
Numerical Instabilities and Resonances
483(3)
Extension to Variable Step Sizes
486(4)
Exercises
490(3)
Bibliography 493(16)
Index 509