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E-raamat: Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems

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Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic ProblemsSuurem pilt
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"Whatever regrets may be, we have done our best." (Sir Ernest Shack­ 0 leton, turning back on 9 January 1909 at 88 23' South.) Brahms struggled for 20 years to write his first symphony. Compared to this, the 10 years we have been working on these two volumes may even appear short. This second volume treats stiff differential equations and differential algebraic equations. It contains three chapters: Chapter IV on one-step (Runge-Kutta) meth­ ods for stiff problems, Chapter V on multistep methods for stiff problems, and Chapter VI on singular perturbation and differential-algebraic equations. Each chapter is divided into sections. Usually the first sections of a chapter are of an introductory nature, explain numerical phenomena and exhibit numerical results. Investigations of a more theoretical nature are presented in the later sections of each chapter. As in Volume I, the formulas, theorems, tables and figures are numbered con­ secutively in each section and indicate, in addition, the section number. In cross references to other chapters the (latin) chapter number is put first. References to the bibliography are again by "author" plus "year" in parentheses. The bibliography again contains only those papers which are discussed in the text and is in no way meant to be complete.

Arvustused

From the reviews of the second edition:

This is a superb book...Throughout, illuminating graphics, sketches and quotes from papers of researchers in the field add an element of easy informality and motivate the text." Mathematics Today

This volume, on nonstiff equations, is the second of a two-volume set. This second volume treats stiff differential equations and differential-algebraic equations. This book is highly recommended as a text for courses in numerical methods for ordinary differential equations and as a reference for the worker. It should be in every library, both academic and industrial. (Teodora-Liliana Rdulescu, Zentralblatt MATH, Vol. 1192, 2010)

Muu info

Second revised and expanded edition
Stiff Problems -- One-Step Methods
Examples of Stiff Equations
2(13)
Chemical Reaction systems
3(1)
Electrical Circuits
4(2)
Diffusion
6(2)
A ``Stiff'' Beam
8(3)
High Oscillations
11(1)
Exercises
11(4)
Stability Analysis for Explicit RK Methods
15(25)
Stability Analysis for Euler's Method
15(1)
Explicit Runge-Kutta Methods
16(2)
Extrapolation Methods
18(1)
Analysis of the Examples of IV.1
18(3)
Automatic Stiffness Detection
21(3)
Step-Control Stability
24(4)
A PI Step Size Control
28(3)
Stabilized Explicit Runge-Kutta Methods
31(6)
Exercises
37(3)
Stability Function of Implicit RK-Methods
40(11)
The Stability Function
40(2)
A-Stability
42(2)
L-Stability and A(αa)-Stability
44(2)
Numerical Results
46(1)
Stability Functions of Order ≥ s
47(1)
Pade Approximations to the Exponential Function
48(1)
Exercises
49(2)
Order Stars
51(20)
Introduction
51(5)
Order and Stability for Rational Approximations
56(2)
Stability of Pade Approximations
58(1)
Comparing Stability Domains
58(3)
Rational Approximations with Real Poles
61(1)
The Real-Pole Sandwich
62(5)
Multiple Real-Pole Approximations
67(3)
Exercises
70(1)
Construction of Implicit Runge-Kutta Methods
71(20)
Gauss Methods
71(1)
Radau IA and Radau IIA Methods
72(3)
Lobatto IIIA, IIIB and IIIC Methods
75(2)
The W -Transformation
77(6)
Construction of Implicit Runge-Kutta Methods
83(1)
Stability Function
84(2)
Positive Functions
86(3)
Exercises
89(2)
Diagonally Implicit RK Methods
91(11)
Order Conditions
91(1)
Stiffy Accurate SDIRK Methods
92(4)
The Stability Function
96(2)
Multiple Real-Pole Approximations with R()=0
98(1)
Choice of Method
99(1)
Exercises
100(2)
Rosenbrock-Type Methods
102(16)
Derivation of the Method
102(2)
Order Conditions
104(4)
The Stability Function
108(1)
Construction of Methods of Order 4
108(3)
Higher Order Methods
111(1)
Implementation of Rosenbrock-Type Methods
111(2)
The ``Hump''
113(1)
Methods with Inexact Jacobian (W -Methods)
114(3)
Exercises
117(1)
Implementation of Implicit Runge-Kutta Methods
118(13)
Reformulation of the Nonlinear System
118(1)
Simplified Newton Iterations
119(2)
The Linear System
121(2)
Step Size Selection
123(4)
Implicit Differential Equations
127(1)
An SDIRK-Code
128(1)
SIRK-Methods
128(2)
Exercises
130(1)
Extrapolation Methods
131(12)
Extrapolation of Symmetric Methods
131(2)
Smoothing
133(1)
The Linearly Implicit Mid-Point Rule
134(4)
The Implicit and Linearly Implicit Euler Methods
138(1)
Implementation
139(3)
Exercises
142(1)
Numerical Experiments
143(24)
The Codes Used
143(1)
Twelve Test Problems
144(8)
Results and Discussion
152(8)
Partitioning and Projection Methods
160(5)
Exercises
165(2)
Contractivity for Linear Problems
167(13)
Euclidean Norms (Theorem of von Neumann)
168(1)
Error Growth Function for Linear Problems
169(3)
Small Nonlinear Perturbations
172(3)
Contractivity in ∥.∥ ∞ and ∥.∥1
175(1)
Study of the Threshold Factor
176(2)
Absolutely Monotonic Functions
178(1)
Exercises
179(1)
B-Stability and Contractivity
180(21)
One-Sided Lipschitz Condition
180(1)
B-Stability and Algebraic Stability
181(2)
Some Algebraically Stable IRK Methods
183(1)
AN -Stability
184(3)
Reducible Runge-Kutta Methods
187(1)
The Equivalence Theorem for S -Irreducible Methods
188(5)
Error Growth Function
193(2)
Computation of γaB(x)
195(4)
Exercises
199(2)
Positive Quadrature Formulas and B-Stable RK-Methods
201(14)
Quadrature Formulas and Related Continued Fractions
201(2)
Number of Positive Weights
203(2)
Characterization of Positive Quadrature Formulas
205(1)
Necessary Conditions for Algebraic Stability
206(3)
Characterization of Algebraically Stable Methods
209(2)
The ``Equivalence'' of A - and B -Stability
211(2)
Exercises
213(2)
Existence and Uniqueness of IRK Solutions
215(10)
Existence
215(2)
A Counterexample
217(1)
Influence of Perturbations and Uniqueness
218(2)
Computation of α0(A1)
220(2)
Methods with Singular A
222(1)
Lobatto IIIC Methods
223(1)
Exercises
223(2)
B-Convergence
225(15)
The Order Reduction Phenomenon
225(3)
The Local Error
228(1)
Error Propagation
229(1)
B-Convergence for Variable Step Sizes
230(2)
B-Convergence Implies Algebraic Stability
232(2)
The Trapezoidal Rule
234(2)
Order Reduction for Rosenbrock Methods
236(1)
Exercises
237(3)
Multistep Methods for Stiff Problems
Stability of Multistep Methods
240(10)
The Stability Region
240(2)
Adams Methods
242(2)
Predictor-Corrector Schemes
244(1)
Nystrom Methods
245(1)
BDF
246(1)
The Second Dahlquist Barrier
247(2)
Exercises
249(1)
``Nearly'' A-Stable Multistep Methods
250(11)
A(αa)-Stability and Stiff Stability
250(1)
High Order A(αa)-Stable Methods
251(2)
Approximating Low Order Methods with High Order Ones
253(1)
A Disc Theorem
254(1)
Accuracy Barriers for Linear Multistep Methods
254(5)
Exercises
259(2)
Generalized Multistep Methods
261(18)
Second Derivative Multistep Methods of Enright
261(4)
Second Derivative BDF Methods
265(1)
Blended Multistep Methods
266(1)
Extended Multistep Methods of Cash
267(3)
Multistep Collocation Methods
270(3)
Methods of ``Radau'' Type
273(2)
Exercises
275(4)
Order Stars on Riemann Surfaces
279(21)
Riemann Surfaces
279(4)
Poles Representing Numerical Work
283(1)
Order and Order Stars
284(2)
The ``Daniel and Moore Conjecture''
286(2)
Methods with Property C
288(2)
General Linear Methods
290(5)
Dual Order Stars
295(2)
Exercises
297(3)
Experiments with Multistep Codes
300(5)
The Codes Used
300(4)
Exercises
304(1)
One-Leg Methods and G-Stability
305(16)
One-Leg (Multistep) Methods
305(1)
Existence and Uniqueness
306(1)
G-Stability
307(2)
An Algebraic Criterion
309(1)
The Equivalence of A-Stability and G-Stability
310(3)
A Criterion for Positive Functions
313(1)
Error Bounds for One-Leg Methods
314(3)
Convergence of A-Stable Multistep Methods
317(2)
Exercises
319(2)
Convergence for Linear Problems
321(18)
Difference Equations for the Global Error
321(2)
The Kreiss Matrix Theorem
323(3)
Some Applications of the Kreiss Matrix Theorem
326(2)
Global Error for Prothero and Robinson Problem
328(1)
Convergence for Linear Systems with Constant Coefficients
329(1)
Matrix Valued Theorem of von Neumann
330(2)
Discrete Variation of Constants Formula
332(5)
Exercises
337(2)
Convergence for Nonlinear Problems
339(17)
Problems Satisfying a One-Sided Lipschitz Condition
339(3)
Multiplier Technique
342(4)
Multipliers and Nonlinearities
346(2)
Discrete Variation of Constants and Perturbations
348(1)
Convergence for Nonlinear Parabolic Problems
349(5)
Exercises
354(2)
Algebraic Stability of General Linear Methods
356(16)
G-Stability
356(1)
Algebraic Stability
357(2)
AN-Stability and Equivalence Results
359(3)
Multistep Runge-Kutta Methods
362(1)
Simplifying Assumptions
363(2)
Quadrature Formulas
365(1)
Algebraically Stable Methods of Order 2s
366(2)
B-Convergence
368(2)
Exercises
370(2)
Singular Perturbation Problems and Index 1 Problems
Solving Index 1 Problems
372(10)
Asymptotic Solution of van der Pol's Equations
372(2)
The ε-Embedding Method for Problems of Index 1
374(1)
State Space Form Method
375(1)
A Transistor Amplifier
376(2)
Problems of the Form Mu=γ(u)
378(2)
Convergence of Runge-Kutta Methods
380(1)
Exercises
381(1)
Multistep Methods
382(6)
Methods for Index 1 Problems
382(1)
Convergence for Singular Perturbation Problems
383(4)
Exercises
387(1)
Epsilon Expansions for Exact and RK Solutions
388(19)
Expansion of the Smooth Solution
388(1)
Expansions with Boundary Layer Terms
389(2)
Estimation of the Remainder
391(1)
Expansion of the Runge-Kutta Solution
392(2)
Convergence of RK-Methods for Differential-Algebraic Systems
394(3)
Existence and Uniqueness of the Runge-Kutta Solution
397(1)
Influence of Perturbations
398(1)
Estimation of the Remainder in the Numerical Solution
399(4)
Numerical Confirmation
403(2)
Perturbed Initial Values
405(1)
Exercises
406(1)
Rosenbrock Methods
407(19)
Definition of the Method
407(1)
Derivatives of the Exact Solution
408(1)
Trees and Elementary Differentials
409(2)
Taylor Expansion of the Exact Solution
411(1)
Taylor Expansion of the Numerical Solution
412(3)
Order Conditions
415(1)
Convergence
416(2)
Stiffly Accurate Rosenbrock Methods
418(2)
Construction of RODAS, a Stiffly Accurate Embedded Method
420(2)
Inconsistent Initial Values
422(2)
Exercises
424(2)
Extrapolation Methods
426(16)
Linearly Implicit Euler Discretization
426(2)
Perturbed Asymptotic Expansion
428(3)
Order Tableau
431(2)
Error Expansion for Singular Perturbation Problems
433(5)
Dense Output
438(3)
Exercises
441(1)
Quasilinear Problems
442(10)
Example: Moving Finite Elements
442(3)
Problems of Index One
445(1)
Numerical Treatment of C(y)y' = f(y)
446(1)
Extrapolation Methods
447(1)
Exercises
448(4)
Differential-Algebraic Equations of Higher Index
The Index and Various Examples
452(16)
Linear Equations with Constants Coefficients
452(2)
Differentiation Index
454(3)
Differential Equations on Manifolds
457(2)
The Perturbation Index
459(2)
Control Problems
461(2)
Mechanical Systems
463(2)
Exercises
465(3)
Index Reduction Methods
468(13)
Index Reduction by Differentiation
468(2)
Stabilization by Projection
470(2)
Differential Equations with Invariants
472(2)
Methods Based on Local State Space Forms
474(3)
Overdetermined Differential-Algebraic Equations
477(1)
Unstructured Higher Index Problems
478(2)
Exercises
480(1)
Multistep Methods for Index 2 DAE
481(11)
Existence and Uniqueness of Numerical Solution
482(2)
Influence of Perturbations
484(1)
The Local Error
485(1)
Convergence for BDF
486(3)
General Multistep Methods
489(1)
Solution of the Nonlinear System by Simplified Newton
490(1)
Exercises
491(1)
Runge-Kutta Methods for Index 2 DAE
492(14)
The Nonlinear System
492(2)
Estimation of the Local Error
494(2)
Convergence for the y-Component
496(1)
Convergence for the z-Component
497(1)
Collocation Methods
498(2)
Superconvergence of Collocation Methods
500(2)
Projected Runge-Kutta Methods
502(2)
Summary of Convergence Results
504(1)
Exercises
505(1)
Order Conditions for Index 2 DAE
506(13)
Derivatives of the Exact Solution
506(1)
Trees and Elementary Differentials
507(1)
Taylor Expansion of the Exact Solution
508(2)
Derivatives of the Numerical Solution
510(2)
Order Conditions
512(2)
Simplifying Assumptions
514(1)
Projected Runge-Kutta Methods
515(3)
Exercises
518(1)
Half-Explicit Methods for Index 2 Systems
519(11)
Half-Explicit Runge-Kutta Methods
520(5)
Extrapolation Methods
525(2)
βB-Blocked Multistep Methods
527(2)
Exercises
529(1)
Computation of Multibody Mechanisms
530(13)
Description of the Model
530(3)
Fortran Subroutines
533(2)
Computation of Consistent Initial Values
535(1)
Numerical Computations
536(5)
A Stiff Mechanical System
541(1)
Exercises
542(1)
Symplectic Methods for Constrained Hamiltonian Systems
543(22)
Properties of the Exact Flow
544(1)
First Order Symplectic Method
545(3)
SHAKE and RATTLE
548(2)
The Lobatto IIIA-IIIB Pair
550(4)
Composition Methods
554(1)
Backward Error Analysis (for ODEs)
555(4)
Backward Error Analysis on Manifolds
559(3)
Exercises
562(3)
Appendix. Fortran Codes 565(12)
Driver for the Code RADAU5
566(2)
Subroutine RADAU5
568(6)
Subroutine RADAUP
574(1)
Subroutine RODAS
574(1)
Subroutine SEULEX
575(1)
Problems with Special Structure
575(1)
Use of SOLOUT and of Dense Output
576(1)
Bibliography 577(28)
Symbol Index 605(2)
Subject Index 607


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