| Titre de série : |
Solving ordinary differential equations, 1 |
| Titre : |
1, Nonstiff problems |
| Type de document : |
Monographie |
| Auteurs : |
E. Hairer, Auteur ; S.P. Norsett, Auteur ; G. Wanner, Auteur |
| Mention d'édition : |
2 |
| Editeur : |
Berlin, Heidelberg, Vienne, New York, ... : Springer |
| Année de publication : |
2000 |
| Collection : |
Springer series in computational mathematics, ISSN 0179-3632 num. 8 |
| Importance : |
528 p. |
| Format : |
17 x 24 cm |
| ISBN/ISSN/EAN : |
978-3-540-56670-0 |
| Note générale : |
Bibliographie |
| Langues : |
Anglais (eng) |
| Descripteur : |
[Vedettes matières IGN] Analyse numérique
[Termes descripteurs IGN] équation différentielle
[Termes descripteurs IGN] valeur propre
|
| Résumé : |
(Editeur) The subject of the book is the solution of nonstiff ordinary differential equations. The first chapter describes the historical development of the classical theory from Newton, Leibniz, Euler and Hamilton to limit cycles and strange attractors. In a second chapter a modern treatment of RungeKutta and extrapolation methods is given. Also included are continuous methods for dense output, parallel RungeKutta methods, special methods for Hamiltonian systems, second order differential equations and delay equations. The third chapter begins with the classical theory of multistep methods, treats variable step size and Nordsieck methods, and concludes with the theory of general linear methods. Many applications from physics, chemistry, biology and astronomy together with computer programs and numerical comparisons are presented. For this second edition many sections have been rewritten and new material included. |
| Note de contenu : |
Chapter 1. Classical Mathematical Theory
1.1 Terminology
1.2 The Oldest Differential Equations
Newton
Leibniz and the Bernoulli Brothers
Variational Calculus
Clairaut
Exercises
1.3 Elementary Integration Methods
First Order Equations
Second Order Equations
Exercises
1.4 Linear Differential Equations
Equations with Constant Coefficient
Variation of Constants
Exercises
1.5 Equations with Weak Singularities
Linear Equations
Nonlinear Equations
Exercises
1.6 Systems of Equations
The Vibrating String and Propagation of sound
Fourier
Lagrangian Mechanics
Hamiltonian Mechanics
Exercises
1.7 A General Existence Theorem
Convergence of Euler's Method
Existence Theorem of Peano
Exercises
1.8 Existence Theory using Iteration Methods and Taylor Series
PicardLindelof Iteration
Taylor Series
Recursive Computation of Taylor Coefficients
Exercise
1.9 Existence Theory for Systems of Equations
Vector Notation
Subordinate Matrix Norms
Exercises
1.10 Differential Inequalities
Introduction
The Fundamental Theorems
Estimates Using OneSided Lipschitz Conditions
Exercises
1.11 Systems of Linear Differential Equations
Resolvent and Wronskian
Inhomogeneous Linear Equations
The AbelLiouvilleJacobiOstrogradskii Identity
Exercises
1.12 Systems with Constant Coefficients
Linearization
Diagonalization
The Schur Decomposition
Numerical Computations
The Jordan Canonical Form
Geometric Representation
Exercises
1.13 Stability
Introduction
The RouthHurwitz Criterion
Computational Considerations
Liapunov Functions
Stability of Nonlinear Systems
Stability of NonAutonomous Systems
Exercises
1.14 Derivatives with Respect to Parameters and Initial Values
The Derivative with Respect to a Parameter
Derivatives with Respect to Initial Values
The Nonlinear VariationofConstants Formula
Flows and VolumePreserving Flows
Canonical Equations and Symplectic Mappings
Exercises
1.15 Boundary Value and Eigenvalue Problems
Boundary Value Problems
SturmLiouville Eigenvalue Problems
Exercises
1.16 Periodic Solutions, Limit Cycles, Strange Attractors
Van der Pol's Equation
Chemical Reactions
Limit Cycles in Higher Dimensions, Hopf Bifurcation
Strange Attractors
The Ups and Downs of the Lorenz Model
Feigenbaum Cascades
Exercises
Chapter 2. RungeKutta and Extrapolation Methods
2.1 The First RungeKutta Methods
General Formulation of RungeKutta Methods
Discussion of Methods of Order 4
"Optimal" Formulas
Numerical Example
Exercises
2.2 Order Conditions for RungeKutta Methods
The Derivatives of the True Solution
Conditions for Order 3
Trees and Elementary Differentials
The Taylor Expansion of the True Solution
Fàà di Bruno's Formula
The Derivatives of the Numerical Solution
The Order Conditions
Exercises
2.3 Error Estimation and Convergence for RK Methods
Rigorous Error Bounds
The Principal Error Term
Estimation of the Global Error
Exercises
2.4 Practical Error Estimation and Step Size Selection .
Richardson Extrapolation
Embedded RungeKutta Formulas
Automatic Step Size Control
Starting Step Size
Numerical Experiments
Exercises
2.5 Explicit RungeKutta Methods of Higher Order
The Butcher Barriers
6Stage, 5th Order Processes
Embedded Formulas of Order 5
Higher Order Processes
Embedded Formulas of High Order
An 8th Order Embedded Method
Exercises
2.6 Dense Output, Discontinuities, Derivatives
Dense Output
Continuous Dormand & Prince Pairs
Dense Output for DOP853
Event Location
Discontinuous Equations
Numerical Computation of Derivatives with Respect to Initial Values and Parameters
Exercises
2.7 Implicit RungeKutta Methods
Existence of a Numerical Solution
The Methods of Kuntzmann and Butcher of Order 2s.
IRK Methods Based on Lobatto Quadrature
Collocation Methods
Exercises
2.8 Asymptotic Expansion of the Global Error
The Global Error
Variable h
Negative h
Properties of the Adjoint Method
Symmetric Methods
Exercises
2.9 Extrapolation Methods
Definition of the Method
The Aitken Neville Algorithm
The Gragg or GBS Method
Asymptotic Expansion for Odd Indices
Existence of Explicit RK Methods of Arbitrary Order
Order and Step Size Control
Dense Output for the GBS Method
Control of the Interpolation Error
Exercises
2.10 Numerical Comparisons
Problems
Performance of the Codes
A "Stretched" Error Estimator for DOP853
Effect of StepNumber Sequence in ODEX
2.11 Parallel Methods
Parallel RungeKutta Methods
Parallel Iterated RungeKutta Methods
Extrapolation Methods
Increasing Reliability
Exercises
2.12 Composition of BSeries
Composition of RungeKutta Methods
BSeries
Order Conditions for RungeKutta Methods
Butcher's “Effective Order”
Exercises
2.13 Higher Derivative Methods
Collocation Methods
HermiteObreschkoff Methods
Fehlberg Methods
General Theory of Order Conditions
Exercises
2.14 Numerical Methods for Second Order Differential Equation
Nyström Methods
The Derivatives of the Exact Solution
The Derivatives of the Numerical Solution
The Order Conditions
On the Construction of Nyström Methods
An Extrapolation Method for y" = f (x, y)
Problems for Numerical Comparisons
Performance of the Codes
Exercises
2.15 PSeries for Partitioned Differential Equations
Derivatives of the Exact Solution, PTrees
PSeries
Order Conditions for Partitioned RungeKutta Methods
Further Applications of PSeries
Exercises
2.16 Symplectic Integration Methods
Symplectic RungeKutta Methods
An Example from Galactic Dynamics
Partitioned RungeKutta Methods
Symplectic Nyström Methods
Conservation of the Hamiltonian; Backward Analysis
Exercises
2.17 Delay Differential Equations
Existence
Constant Step Size Methods for Constant Delay
Variable Step Size Methods
Stability
An Example from Population Dynamics
Infectious Disease Modelling
An Example from Enzyme Kinetics
A Mathematical Model in Immunology
IntegroDifferential Equations
Exercises
Chapter 3. Multistep Methods, and General Linear Methods
3.1 Classical Linear Multistep Formulas
Explicit Adams Methods
Implicit Adams Methods
Numerical Experiment
Explicit Nyström Methods
MilneSimpson Methods
Methods Based on Differentiation (BDF)
Exercises
3.2 Local Error and Order Conditions
Local Error of a Multistep Method
Order of a Multistep Method
Error Constant
Irreducible Methods
The Peano Kernel of a Multistep Method
Exercises
3.3 Stability and the First Dahlquist Barrier
Stability of the BDFFormulas
Highest Attainable Order of Stable Multistep Methods
Exercises
3.4 Convergence of Multistep Methods
Formulation as OneStep Method
Proof of Convergence
Exercises
3.5 Variable Step Size Multistep Methods
Variable Step Size Adams Methods
Recurrence Relations for g;(n), ;(n) and (n)
Variable Step Size BDF
General Variable Step Size Methods and Their Orders
Stability
Convergence
Exercises
3.6 Nordsieck Methods
Equivalence with Multistep Methods
Implicit Adams Methods
BDFMethods
Exercises
3.7 Implementation and Numerical Comparisons
Step Size and Order Selection
Some Available Codes
Numerical Comparisons
3.8 General Linear Methods
A General Integration Procedure
Stability and Order
Convergence
Order Conditions for General Linear Methods
Construction of General Linear Methods
Exercises
3.9 Asymptotic Expansion of the Global Error
An Instructive Example
Asymptotic Expansion for Strictly Stable Methods (8.4)
Weakly Stable Methods
The Adjoint Method
Symmetric Methods
Exercises
3.10 Multistep Methods for Second Order Differential Equations
Explicit Störmer Methods
Implicit Störmer Methods
Numerical Example
General Formulation
Convergence
Asymptotic Formula for the Global Error
Rounding Errors
Exercises
Appendis. Fortran Codes
Driver for the Code DOPRI5
Subroutine DOPRIS
Subroutine DOP853
Subroutine ODEX
Subroutine ODEX2
Driver for the Code RETARD
Subroutine RETARD
Bibliography
Symbol Index
Subject Index |
| Numéro de notice : |
11370A |
| Affiliation des auteurs : |
non IGN |
| Thématique : |
MATHEMATIQUE |
| Nature : |
Monographie |
| Permalink : |
https://documentation.ensg.eu/index.php?lvl=notice_display&id=54396 |
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Solving ordinary differential equations, 1. 1, Nonstiff problems (2000)
