Solving ordinary differential equations: 1 Nonstiff problems
Gespeichert in:
| Format: | Buch |
|---|---|
| Sprache: | English |
| Veröffentlicht: |
Berlin u.a.
Springer
1993
|
| Ausgabe: | 2., rev. ed. |
| Schriftenreihe: | Springer series in computational mathematics
8 |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Literaturverz. S. [491] - 519 |
| Beschreibung: | XV, 528 S. graph. Darst. |
| ISBN: | 3540566708 0387566708 |
Internformat
MARC
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| 245 | 1 | 0 | |a Solving ordinary differential equations |n 1 |p Nonstiff problems |c E. Hairer ; S. P. Nørsett ; G. Wanner |
| 250 | |a 2., rev. ed. | ||
| 264 | 1 | |a Berlin u.a. |b Springer |c 1993 | |
| 300 | |a XV, 528 S. |b 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 8 | |
| 490 | 0 | |a Springer series in computational mathematics |v ... | |
| 500 | |a Literaturverz. S. [491] - 519 | ||
| 700 | 1 | |a Hairer, Ernst |d 1949- |e Sonstige |0 (DE-588)139445188 |4 oth | |
| 700 | 1 | |a Nørsett, Syvert P. |d 1944-2024 |e Sonstige |0 (DE-588)13758718X |4 oth | |
| 700 | 1 | |a Wanner, Gerhard |d 1942- |e Sonstige |0 (DE-588)13944534X |4 oth | |
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| 943 | 1 | |a oai:aleph.bib-bvb.de:BVB01-015219118 | |
Datensatz im Suchindex
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Contents
Chapter I. Classical Mathematical Theory
1.1 Terminology 2
1.2 The Oldest Differential Equations 4
Newton 4
Leibniz and the Bernoulli Brothers 6
Variational Calculus 7
Clairaut 9
Exercises 10
1.3 Elementary Integration Methods 12
First Order Equations 12
Second Order Equations 13
Exercises 14
1.4 Linear Differential Equations 16
Equations with Constant Coefficients 16
Variation of Constants 18
Exercises 19
1.5 Equations with Weak Singularities 20
Linear Equations 20
Nonlinear Equations 23
Exercises 24
1.6 Systems of Equations 26
The Vibrating String and Propagation of Sound 26
Fourier 29
Lagrangian Mechanics 30
Hamiltonian Mechanics 32
Exercises 34
1.7 A General Existence Theorem 35
Convergence of Euler's Method 35
Existence Theorem of Peano 41
Exercises 43
1.8 Existence Theory using Iteration Methods and Taylor Series 44
Picard Lindelof Iteration 45
Taylor Series 46
Recursive Computation of Taylor Civflieienls 47
Hxereises 49
X Contents
1.9 Existence Theory for Systems of Equations 51
Vector Notation 52
Subordinate Matrix Norms 53
Exercises 55
1.10 Differential Inequalities 56
Introduction 56
The Fundamental Theorems 57
Estimates Using One Sided Lipschitz Conditions 60
Exercises 62
1.11 Systems of Linear Differential Equations 64
Resolvent and Wronskian 65
Inhomogeneous Linear Equations 66
The Abel Liouville Jacobi Ostrogradskii Identity 66
Exercises 67
1.12 Systems with Constant Coefficients 69
Linearization 69
Diagonalization 69
The Schur Decomposition 70
Numerical Computations 72
The Jordan Canonical Form 73
Geometric Representation 77
Exercises 78
1.13 Stability 80
Introduction 80
The Routh Hurwitz Criterion 81
Computational Considerations 85
Liapunov Functions 86
Stability of Nonlinear Systems 87
Stability of Non Autonomous Systems 88
Exercises 89
1.14 Derivatives with Respect to Parameters and Initial Values . 92
The Derivative with Respect to a Parameter 93
Derivatives with Respect to Initial Values 95
The Nonlinear Variation of Constants Formula 96
Flows and Volume Preserving Flows 97
Canonical Equations and Symplectic Mappings 100
Exercises 104
1.15 Boundary Value and Eigenvalue Problems 105
Boundary Value Problems 105
Sturm Liouville Eigenvalue Problems 107
Exercises 110
1.16 Periodic Solutions, Limit Cycles, Strange Attractors Ill
Van der Pol's Equation Ill
Chemical Reactions 115
Limit Cycles in Higher Dimensions, Hopf Bifurcation 117
Strange Attractors 120
The Ups and Downs of the Lorenz Model 123
Feigenbaum Cascades 124
Exercises 126
Contents XI
Chapter II. Runge Kutta and Extrapolation Methods
11.1 The First Runge Kutta Methods 132
General Formulation of Runge Kutta Methods 134
Discussion of Methods of Order 4 135
"Optimal" Formulas 139
Numerical Example 140
Exercises 141
11.2 Order Conditions for Runge Kutta Methods 143
The Derivatives of the True Solution 145
Conditions for Order 3 145
Trees and Elementary Differentials 145
The Taylor Expansion of the True Solution 148
Faa di Bruno's Formula 149
The Derivatives of the Numerical Solution 151
The Order Conditions 153
Exercises 154
11.3 Error Estimation and Convergence for RK Methods 156
Rigorous Error Bounds 156
The Principal Error Term 158
Estimation of the Global Error 159
Exercises 163
11.4 Practical Error Estimation and Step Size Selection 164
Richardson Extrapolation 164
Embedded Runge Kutta Formulas 165
Automatic Step Size Control 167
Starting Step Size 169
Numerical Experiments 170
Exercises 172
11.5 Explicit Runge Kutta Methods of Higher Order 173
The Butcher Barriers 173
6 Stage, 5th Order Processes 175
Embedded Formulas of Order 5 176
Higher Order Processes 179
Embedded Formulas of High Order 180
An 8th Order Embedded Method 181
Exercises 185
11.6 Dense Output, Discontinuities, Derivatives 188
Dense Output 188
Continuous Dormand Prince Pairs 191
Dense Output for DOP853 194
Event Location 195
Discontinuous Equations 196
Numerical Computation of Derivatives with Respect
to Initial Values and Parameters 200
Exercises 202
11.7 Implicit Runge Kutta Methods 204
Existence of a Numerical Solution 206
The Methods of Kuntzmann and Butcher of Order 2s 208
IRK Methods Based on Lobatto Quadrature 210
XII Contents
Collocation Methods 211
Exercises 214
11.8 Asymptotic Expansion of the Global Error 216
The Global Error 216
Variable h 218
Negative h 219
Properties of the Adjoint Method 220
Symmetric Methods 221
Exercises 223
11.9 Extrapolation Methods 224
Definition of the Method 224
The Aitken Neville Algorithm 226
The Gragg or GBS Method 228
Asymptotic Expansion for Odd Indices 231
Existence of Explicit RK Methods of Arbitrary Order 232
Order and Step Size Control 233
Dense Output for the GBS Method 237
Control of the Interpolation Error 240
Exercises 241
11.10 Numerical Comparisons 244
Problems 244
Performance of the Codes 249
A "Stretched" Error Estimator for DOP853 254
Effect of Step Number Sequence in ODEX 256
11.11 Parallel Methods 257
Parallel Runge Kutta Methods 258
Parallel Iterated Runge Kutta Methods 259
Extrapolation Methods 261
Increasing Reliability 261
Exercises 263
11.12 Composition of B Series 264
Composition of Runge Kutta Methods 264
B Series 266
Order Conditions for Runge Kutta Methods 269
Butcher's "Effective Order" 270
Exercises 272
11.13 Higher Derivative Methods 274
Collocation Methods 275
Hermite Obreschkoff Methods 277
Fehlberg Methods 278
General Theory of Order Conditions 280
Exercises 281
11.14 Numerical Methods for Second Order Differential Equations 283
Nystrdm Methods 284
The Derivatives of the Exact Solution 286
The Derivatives of the Numerical Solution 288
The Order Conditions 290
On the Construction of Nystrom Methods 291
An Extrapolation Method for y" — fix, y) 294
Problems for Numerical Comparisons 296
Contents XIII
Performance of the Codes 298
Exercises 300
11.15 P Series for Partitioned Differential Equations 302
Derivatives of the Exact Solution, P Trees 303
P Series 306
Order Conditions for Partitioned Runge Kutta Methods 307
Further Applications of P Series 308
Exercises 311
11.16 Symplectic Integration Methods 312
Symplectic Runge Kutta Methods 315
An Example from Galactic Dynamics 319
Partitioned Runge Kutta Methods 326
Symplectic Nystrom Methods 330
Conservation of the Hamiltonian; Backward Analysis 333
Exercises 337
11.17 Delay Differential Equations 339
Existence 339
Constant Step Size Methods for Constant Delay 341
Variable Step Size Methods 342
Stability 343
An Example from Population Dynamics 345
Infectious Disease Modelling 347
An Example from Enzyme Kinetics 248
A Mathematical Model in Immunology 349
Integro Differential Equations 351
Exercises 352
Chapter III. Multistep Methods
and General Linear Methods
111.1 Classical Linear Multistep Formulas 356
Explicit Adams Methods 357
Implicit Adams Methods 359
Numerical Experiment 361
Explicit Nystrom Methods 362
Milne Simpson Methods 363
Methods Based on Differentiation (BDF) 364
Exercises 366
111.2 Local Error and Order Conditions 368
Local Error of a Multistep Method 368
Order of a Multistep Method 370
Error Constant 372
Irreducible Methods 374
The Peano Kernel of a Multistep Method 375
Exercises 377
III3 Stability and the First Dahlquist Barrier 378
Stability of the BDF Formulas 380
Highest Attainable Order of Stable Multistep Methods 383
Exercises 387
XIV Contents
111.4 Convergence of Multistep Methods 391
Formulation as One Step Method 393
Proof of Convergence 395
Exercises 396
111.5 Variable Step Size Multistep Methods 397
Variable Step Size Adams Methods 397
Recurrence Relations for gj(n), $j(n) and $|(n) 399
Variable Step Size BDF 400
General Variable Step Size Methods and Their Orders 401
Stability 402
Convergence 407
Exercises 409
111.6 Nordsieck Methods 410
Equivalence with Multistep Methods 412
Implicit Adams Methods 417
BDF Methods 419
Exercises 420
111.7 Implementation and Numerical Comparisons 421
Step Size and Order Selection 421
Some Available Codes 423
Numerical Comparisons 427
111.8 General Linear Methods 430
A General Integration Procedure 431
Stability and Order 436
Convergence 438
Order Conditions for General Linear Methods 441
Construction of General Linear Methods 443
Exercises 445
IIL9 Asymptotic Expansion of the Global Error 448
An Instructive Example 448
Asymptotic Expansion for Strictly Stable Methods (8.4) 450
Weakly Stable Methods 454
The Adjoint Method 457
Symmetric Methods 459
Exercises 460
111.10 Multistep Methods for Second Order Differential Equations 461
Explicit Stormer Methods 462
Implicit Stormer Methods 464
Numerical Example 465
General Formulation 467
Convergence 468
Asymptotic Formula for the Global Error 471
Rounding Errors 472
Exercises 473
Appendix. Fortran Codes 475
Driver for the Code DOPRI5 475
Subroutine DOPRI5 477
Subroutine DOP853 481
Subroutine ODEX 482
Contents XV
Subroutine 0DEX2 484
Driver for the Code RETARD 486
Subroutine RETARD 488
Bibliography 491
Symbol Index 521
Subject Index 523 |
| adam_txt |
Contents
Chapter I. Classical Mathematical Theory
1.1 Terminology 2
1.2 The Oldest Differential Equations 4
Newton 4
Leibniz and the Bernoulli Brothers 6
Variational Calculus 7
Clairaut 9
Exercises 10
1.3 Elementary Integration Methods 12
First Order Equations 12
Second Order Equations 13
Exercises 14
1.4 Linear Differential Equations 16
Equations with Constant Coefficients 16
Variation of Constants 18
Exercises 19
1.5 Equations with Weak Singularities 20
Linear Equations 20
Nonlinear Equations 23
Exercises 24
1.6 Systems of Equations 26
The Vibrating String and Propagation of Sound 26
Fourier 29
Lagrangian Mechanics 30
Hamiltonian Mechanics 32
Exercises 34
1.7 A General Existence Theorem 35
Convergence of Euler's Method 35
Existence Theorem of Peano 41
Exercises 43
1.8 Existence Theory using Iteration Methods and Taylor Series 44
Picard Lindelof Iteration 45
Taylor Series 46
Recursive Computation of Taylor Civflieienls 47
Hxereises 49
X Contents
1.9 Existence Theory for Systems of Equations 51
Vector Notation 52
Subordinate Matrix Norms 53
Exercises 55
1.10 Differential Inequalities 56
Introduction 56
The Fundamental Theorems 57
Estimates Using One Sided Lipschitz Conditions 60
Exercises 62
1.11 Systems of Linear Differential Equations 64
Resolvent and Wronskian 65
Inhomogeneous Linear Equations 66
The Abel Liouville Jacobi Ostrogradskii Identity 66
Exercises 67
1.12 Systems with Constant Coefficients 69
Linearization 69
Diagonalization 69
The Schur Decomposition 70
Numerical Computations 72
The Jordan Canonical Form 73
Geometric Representation 77
Exercises 78
1.13 Stability 80
Introduction 80
The Routh Hurwitz Criterion 81
Computational Considerations 85
Liapunov Functions 86
Stability of Nonlinear Systems 87
Stability of Non Autonomous Systems 88
Exercises 89
1.14 Derivatives with Respect to Parameters and Initial Values . 92
The Derivative with Respect to a Parameter 93
Derivatives with Respect to Initial Values 95
The Nonlinear Variation of Constants Formula 96
Flows and Volume Preserving Flows 97
Canonical Equations and Symplectic Mappings 100
Exercises 104
1.15 Boundary Value and Eigenvalue Problems 105
Boundary Value Problems 105
Sturm Liouville Eigenvalue Problems 107
Exercises 110
1.16 Periodic Solutions, Limit Cycles, Strange Attractors Ill
Van der Pol's Equation Ill
Chemical Reactions 115
Limit Cycles in Higher Dimensions, Hopf Bifurcation 117
Strange Attractors 120
The Ups and Downs of the Lorenz Model 123
Feigenbaum Cascades 124
Exercises 126
Contents XI
Chapter II. Runge Kutta and Extrapolation Methods
11.1 The First Runge Kutta Methods 132
General Formulation of Runge Kutta Methods 134
Discussion of Methods of Order 4 135
"Optimal" Formulas 139
Numerical Example 140
Exercises 141
11.2 Order Conditions for Runge Kutta Methods 143
The Derivatives of the True Solution 145
Conditions for Order 3 145
Trees and Elementary Differentials 145
The Taylor Expansion of the True Solution 148
Faa di Bruno's Formula 149
The Derivatives of the Numerical Solution 151
The Order Conditions 153
Exercises 154
11.3 Error Estimation and Convergence for RK Methods 156
Rigorous Error Bounds 156
The Principal Error Term 158
Estimation of the Global Error 159
Exercises 163
11.4 Practical Error Estimation and Step Size Selection 164
Richardson Extrapolation 164
Embedded Runge Kutta Formulas 165
Automatic Step Size Control 167
Starting Step Size 169
Numerical Experiments 170
Exercises 172
11.5 Explicit Runge Kutta Methods of Higher Order 173
The Butcher Barriers 173
6 Stage, 5th Order Processes 175
Embedded Formulas of Order 5 176
Higher Order Processes 179
Embedded Formulas of High Order 180
An 8th Order Embedded Method 181
Exercises 185
11.6 Dense Output, Discontinuities, Derivatives 188
Dense Output 188
Continuous Dormand Prince Pairs 191
Dense Output for DOP853 194
Event Location 195
Discontinuous Equations 196
Numerical Computation of Derivatives with Respect
to Initial Values and Parameters 200
Exercises 202
11.7 Implicit Runge Kutta Methods 204
Existence of a Numerical Solution 206
The Methods of Kuntzmann and Butcher of Order 2s 208
IRK Methods Based on Lobatto Quadrature 210
XII Contents
Collocation Methods 211
Exercises 214
11.8 Asymptotic Expansion of the Global Error 216
The Global Error 216
Variable h 218
Negative h 219
Properties of the Adjoint Method 220
Symmetric Methods 221
Exercises 223
11.9 Extrapolation Methods 224
Definition of the Method 224
The Aitken Neville Algorithm 226
The Gragg or GBS Method 228
Asymptotic Expansion for Odd Indices 231
Existence of Explicit RK Methods of Arbitrary Order 232
Order and Step Size Control 233
Dense Output for the GBS Method 237
Control of the Interpolation Error 240
Exercises 241
11.10 Numerical Comparisons 244
Problems 244
Performance of the Codes 249
A "Stretched" Error Estimator for DOP853 254
Effect of Step Number Sequence in ODEX 256
11.11 Parallel Methods 257
Parallel Runge Kutta Methods 258
Parallel Iterated Runge Kutta Methods 259
Extrapolation Methods 261
Increasing Reliability 261
Exercises 263
11.12 Composition of B Series 264
Composition of Runge Kutta Methods 264
B Series 266
Order Conditions for Runge Kutta Methods 269
Butcher's "Effective Order" 270
Exercises 272
11.13 Higher Derivative Methods 274
Collocation Methods 275
Hermite Obreschkoff Methods 277
Fehlberg Methods 278
General Theory of Order Conditions 280
Exercises 281
11.14 Numerical Methods for Second Order Differential Equations 283
Nystrdm Methods 284
The Derivatives of the Exact Solution 286
The Derivatives of the Numerical Solution 288
The Order Conditions 290
On the Construction of Nystrom Methods 291
An Extrapolation Method for y" — fix, y) 294
Problems for Numerical Comparisons 296
Contents XIII
Performance of the Codes 298
Exercises 300
11.15 P Series for Partitioned Differential Equations 302
Derivatives of the Exact Solution, P Trees 303
P Series 306
Order Conditions for Partitioned Runge Kutta Methods 307
Further Applications of P Series 308
Exercises 311
11.16 Symplectic Integration Methods 312
Symplectic Runge Kutta Methods 315
An Example from Galactic Dynamics 319
Partitioned Runge Kutta Methods 326
Symplectic Nystrom Methods 330
Conservation of the Hamiltonian; Backward Analysis 333
Exercises 337
11.17 Delay Differential Equations 339
Existence 339
Constant Step Size Methods for Constant Delay 341
Variable Step Size Methods 342
Stability 343
An Example from Population Dynamics 345
Infectious Disease Modelling 347
An Example from Enzyme Kinetics 248
A Mathematical Model in Immunology 349
Integro Differential Equations 351
Exercises 352
Chapter III. Multistep Methods
and General Linear Methods
111.1 Classical Linear Multistep Formulas 356
Explicit Adams Methods 357
Implicit Adams Methods 359
Numerical Experiment 361
Explicit Nystrom Methods 362
Milne Simpson Methods 363
Methods Based on Differentiation (BDF) 364
Exercises 366
111.2 Local Error and Order Conditions 368
Local Error of a Multistep Method 368
Order of a Multistep Method 370
Error Constant 372
Irreducible Methods 374
The Peano Kernel of a Multistep Method 375
Exercises 377
III3 Stability and the First Dahlquist Barrier 378
Stability of the BDF Formulas 380
Highest Attainable Order of Stable Multistep Methods 383
Exercises 387
XIV Contents
111.4 Convergence of Multistep Methods 391
Formulation as One Step Method 393
Proof of Convergence 395
Exercises 396
111.5 Variable Step Size Multistep Methods 397
Variable Step Size Adams Methods 397
Recurrence Relations for gj(n), $j(n) and $|(n) 399
Variable Step Size BDF 400
General Variable Step Size Methods and Their Orders 401
Stability 402
Convergence 407
Exercises 409
111.6 Nordsieck Methods 410
Equivalence with Multistep Methods 412
Implicit Adams Methods 417
BDF Methods 419
Exercises 420
111.7 Implementation and Numerical Comparisons 421
Step Size and Order Selection 421
Some Available Codes 423
Numerical Comparisons 427
111.8 General Linear Methods 430
A General Integration Procedure 431
Stability and Order 436
Convergence 438
Order Conditions for General Linear Methods 441
Construction of General Linear Methods 443
Exercises 445
IIL9 Asymptotic Expansion of the Global Error 448
An Instructive Example 448
Asymptotic Expansion for Strictly Stable Methods (8.4) 450
Weakly Stable Methods 454
The Adjoint Method 457
Symmetric Methods 459
Exercises 460
111.10 Multistep Methods for Second Order Differential Equations 461
Explicit Stormer Methods 462
Implicit Stormer Methods 464
Numerical Example 465
General Formulation 467
Convergence 468
Asymptotic Formula for the Global Error 471
Rounding Errors 472
Exercises 473
Appendix. Fortran Codes 475
Driver for the Code DOPRI5 475
Subroutine DOPRI5 477
Subroutine DOP853 481
Subroutine ODEX 482
Contents XV
Subroutine 0DEX2 484
Driver for the Code RETARD 486
Subroutine RETARD 488
Bibliography 491
Symbol Index 521
Subject Index 523 |
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| id | DE-604.BV022004479 |
| illustrated | Illustrated |
| index_date | 2024-07-02T16:11:37Z |
| indexdate | 2024-09-06T00:29:07Z |
| institution | BVB |
| isbn | 3540566708 0387566708 |
| language | English |
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| physical | XV, 528 S. graph. Darst. |
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| publisher | Springer |
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| series | Springer series in computational mathematics |
| series2 | Springer series in computational mathematics |
| spelling | Solving ordinary differential equations 1 Nonstiff problems E. Hairer ; S. P. Nørsett ; G. Wanner 2., rev. ed. Berlin u.a. Springer 1993 XV, 528 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Springer series in computational mathematics 8 Springer series in computational mathematics ... Literaturverz. S. [491] - 519 Hairer, Ernst 1949- Sonstige (DE-588)139445188 oth Nørsett, Syvert P. 1944-2024 Sonstige (DE-588)13758718X oth Wanner, Gerhard 1942- Sonstige (DE-588)13944534X oth (DE-604)BV000612774 1 Springer series in computational mathematics 8 (DE-604)BV000012004 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015219118&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Solving ordinary differential equations Springer series in computational mathematics |
| title | Solving ordinary differential equations |
| title_auth | Solving ordinary differential equations |
| title_exact_search | Solving ordinary differential equations |
| title_exact_search_txtP | Solving ordinary differential equations |
| title_full | Solving ordinary differential equations 1 Nonstiff problems E. Hairer ; S. P. Nørsett ; G. Wanner |
| title_fullStr | Solving ordinary differential equations 1 Nonstiff problems E. Hairer ; S. P. Nørsett ; G. Wanner |
| title_full_unstemmed | Solving ordinary differential equations 1 Nonstiff problems E. Hairer ; S. P. Nørsett ; G. Wanner |
| title_short | Solving ordinary differential equations |
| title_sort | solving ordinary differential equations nonstiff problems |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015219118&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000612774 (DE-604)BV000012004 |
| work_keys_str_mv | AT hairerernst solvingordinarydifferentialequations1 AT nørsettsyvertp solvingordinarydifferentialequations1 AT wannergerhard solvingordinarydifferentialequations1 |