ISBN: 3-540-56670-8
TITLE: Solving Ordinary Differential Equations I
AUTHOR: Hairer, Ernst; Norsett, Syvert P.; Wanner, Gerhard
TOC:

Chapter I. Classical Mathematical Theory 
I.1 Terminology 2 
I.2 The Oldest Differential Equations 4 
Newton 4 
Leibniz and the Bernoulli Brothers 6 
Variational Calculus 7 
Clairaut 9 
Exercises 10 
I.3 Elementary Integration Methods 12 
First Order Equations 12 
Second Order Equations 13 
Exercises 14 
I.4 Linear Differential Equations 16 
Equations with Constant Coefficients 16 
Variation of Constants 18 
Exercises 19 
I.5 Equations with Weak Singularities 20 
Linear Equations 20 
Nonlinear Equations 23 
Exercises 24 
I.6 Systems of Equations 26 
The Vibrating String and Propagation of Sound 26 
Fourier 29 
Lagrangian Mechanics 30 
Hamiltonian Mechanics 32 
Exercises 34 
I.7 A General Existence Theorem 35 
Convergence of Euler's Method 35 
Existence Theorem of Peano 41 
Exercises 43 
I.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 
I.9 Existence Theory for Systems of Equations 51 
Vector Notation 52 
Subordinate Matrix Norms 53 
Exercises 55 
I.10 Differential Inequalities 56 
Introduction 56 
The Fundamental Theorems 57 
Estimates Using One-Sided Lipschitz Conditions 60 
Exercises 62 
I.11 Systems of Linear Differential Equations 64 
Resolvent and Wronskian 65 
Inhomogeneous Linear Equations 66 
The Abel-Liouville-Jacobi-Ostrogradskii Identity 66 
Exercises 67 
I.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 
I.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 
I.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 
I.15 Boundary Value and Eigenvalue Problems 105 
Boundary Value Problems 105 
Sturm-Liouville Eigenvalue Problems 107 
Exercises 110 
I.16 Periodic Solutions, Limit Cycles, Strange Attractors 111 
Van der Pol's Equation 111 
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 
Chapter II. Runge-Kutta and Extrapolation Methods 
II.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 
II.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 
II.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 
II.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 
II.5 Explicit Runge-Kutta Methods of Higher Order 173 
The Butcher Barriers 173 
6-Stage, 5 th Order Processes 175 
Embedded Formulas of Order 5 176 
Higher Order Processes 179 
Embedded Formulas of High Order 180 
An 8 th Order Embedded Method 181 
Exercises 185 
II.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 
II.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 
Collocation Methods 211 
Exercises 214 
II.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 
II.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 Implicit 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 
II.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 
II.11 Parallel Methods 257 
Parallel Runge-Kutta Methods 258 
Parallel Iterated Runge-Kutta Methods 259 
Extrapolation Methods 261 
Increasing Reliability 261 
Exercises 263 
II.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 
II.13 Higher Derivative Methods 274 
Collocation Methods 275 
Hermite-Obreschkoff Methods 277 
Fehlherg Methods 278 
General Theory of Order Conditions 280 
Exercises 281 
II.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 
Performance of the Codes 298 
Exercises 300 
II.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 
II.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 
II.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 348 
A Mathematical Model in Immunology 349 
Integro-Differential Equations 351 
Exercises 352 
Chapter III. Multistep Methods 
and General Linear Methods 
III.1 Classical Linear Multistep Formulas 356 
Explicit Adams Methods 357 
Implicit Adams Methods 359 
Numerical Experiment 361 
Explicit Nyström Methods 362 
Milne­Simpson Methods 363 
Methods Based on Differentiation (BDF) 364 
Exercises 366 
III.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 
III.3 Stability and the First Dahlquist Barrier 378 
Stability of the BDE-Formulas 380 
Highest Attainable Order of Stable Multistep Methods 383 
Exercises 387 
III.4 Convergence of Multistep Methods 391 
Formulation as One-Step Method 393 
Proof of Convergence 395 
Exercises 396 
III.5 Variable Step Size Multistep Methods 397 
Variable Step Size Adams Methods 397 
Recurrence Relations for g_j(n), Phi_j(n) and Phi*_j(n) 399 
Variable Step Size BDF 400 
General Variable Step Size Methods and Their Orders 401 
Stability 402 
Convergence 407 
Exercises 409 
III.6 Nordsieck Methods 410 
Equivalence with Multistep Methods 412 
Implicit Adams Methods 417 
BDF-Methods 419 
Exercises 420 
III.7 Implementation and Numerical Comparisons 421 
Step Size and Order Selection 421 
Some Available Codes 423 
Numerical Comparisons 427 
III.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 
III.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 
III.10 Multistep Methods for Second Order Differential Equations 461 
Explicit Störmer Methods 462 
Implicit Störmer 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 
Subroutine ODEX2 484 
Driver for the Code RETARD 486 
Subroutine RETARD 488 
Bibliography 491 
Symbol Index 521 
Subject Index 523 
END
