Holdings: Solving ordinary differential equations

Solving ordinary differential equations: 1 Nonstiff problems

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Bibliographic Details
Format:	Book
Language:	English
Published:	Berlin u.a. Springer 2008
Edition:	2. rev. ed., corr. 3. printing
Series:	Springer series in computational mathematics 8
Subjects:	Differential equations > Numerical solutions
Online Access:	Inhaltsverzeichnis
Item Description:	Hier auch später erschienene, unveränderte Nachdrucke
Physical Description:	XV, 528 S. graph. Darst.
ISBN:	9783540566700 9783642051630

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adam_text	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 . І8 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-Lindelöf Iteration . 45 Taylor Series . 46 Recursive Computation of Taylor Coefficients . 47 Exercises . 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 I.I 1 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 Poľs 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 IL Runge-Kutta and Extrapolation Methods ИЛ 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 Faà 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 П.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 И.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 II.8 Asymptotic Expansion of the Global Error . 216 The Global Error . 216 Variable h . 2l8 Negative h . 219 Properties of the Adjoint Method . 220 Symmetric Methods . 221 Exercises . 223 П.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 Nyström Methods . 284 The Derivatives of the Exact Solution . 286 The Derivatives of the Numerical Solution . 288 The Order Conditions . 290 On the Construction of Nyström Methods . 291 An Extrapolation Method for y" — f(x, 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 Nyström 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 HI.l Classical Linear Multistep Formulas . 356 Explicit Adams Methods . 357 Implicit Adams Methods . 359 Numerical Experiment . 361 Explicit Nyström Methods . 362 Milne-Simpson Methods . 363 Methods Based on Differentiation (BDF) . 364 Exercises . 366 Ш.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 IIL3 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 Ш.4 Convergence of Multistep Methods . 391 Formulation as One-Step Method . 393 Proof of Convergence . 395 Exercises . 3% Ш.5 Variable Step Size Multistep Methods . 397 Variable Step Size Adams Methods . 397 Recurrence Relations for 5j(n), Фј(п) and Фј(п) . 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 Ш.7 Implementation and Numerical Comparisons . 421 Step Size and Order Selection . 421 Some Available Codes . 423 Numerical Comparisons . 427 Ш.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 Ш.9 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 ШЛО Multistep Methods for Second Order Differential Equations 461 Explicit Stornier Methods . 462 Implicit Stornier 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 . І8 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-Lindelöf Iteration . 45 Taylor Series . 46 Recursive Computation of Taylor Coefficients . 47 Exercises . 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 I.I 1 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 Poľs 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 IL Runge-Kutta and Extrapolation Methods ИЛ 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 Faà 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 П.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 И.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 II.8 Asymptotic Expansion of the Global Error . 216 The Global Error . 216 Variable h . 2l8 Negative h . 219 Properties of the Adjoint Method . 220 Symmetric Methods . 221 Exercises . 223 П.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 Nyström Methods . 284 The Derivatives of the Exact Solution . 286 The Derivatives of the Numerical Solution . 288 The Order Conditions . 290 On the Construction of Nyström Methods . 291 An Extrapolation Method for y" — f(x, 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 Nyström 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 HI.l Classical Linear Multistep Formulas . 356 Explicit Adams Methods . 357 Implicit Adams Methods . 359 Numerical Experiment . 361 Explicit Nyström Methods . 362 Milne-Simpson Methods . 363 Methods Based on Differentiation (BDF) . 364 Exercises . 366 Ш.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 IIL3 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 Ш.4 Convergence of Multistep Methods . 391 Formulation as One-Step Method . 393 Proof of Convergence . 395 Exercises . 3% Ш.5 Variable Step Size Multistep Methods . 397 Variable Step Size Adams Methods . 397 Recurrence Relations for 5j(n), Фј(п) and Фј(п) . 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 Ш.7 Implementation and Numerical Comparisons . 421 Step Size and Order Selection . 421 Some Available Codes . 423 Numerical Comparisons . 427 Ш.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 Ш.9 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 ШЛО Multistep Methods for Second Order Differential Equations 461 Explicit Stornier Methods . 462 Implicit Stornier 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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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., corr. 3. printing Berlin u.a. Springer 2008 XV, 528 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Springer series in computational mathematics 8 Springer series in computational mathematics ... Hier auch später erschienene, unveränderte Nachdrucke Differential equations Numerical solutions 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 8 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016990755&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis
spellingShingle	Solving ordinary differential equations Springer series in computational mathematics Differential equations Numerical solutions
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
topic	Differential equations Numerical solutions
topic_facet	Differential equations Numerical solutions
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016990755&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

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