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A new edition of this classic work, comprehensively revised to present exciting new developments in this important subject

The study of numerical methods for solving ordinary differential equations is constantly developing and regenerating, and this third edition of a popular classic volume, written by one of the world's leading experts in the field, presents an account of the subject which reflects both its historical and well-established place in computational science and its vital role as a cornerstone of modern applied mathematics.

In addition to serving as a broad and comprehensive study of numerical methods for initial value problems, this book contains a special emphasis on Runge-Kutta methods by the mathematician who transformed the subject into its modern form dating from his classic 1963 and 1972 papers. A second feature is general linear methods which have now matured and grown from being a framework for a unified theory of a wide range of diverse numerical schemes to a source of new and practical algorithms in their own right. As the founder of general linear method research, John Butcher has been a leading contributor to its development; his special role is reflected in the text. The book is written in the lucid style characteristic of the author, and combines enlightening explanations with rigorous and precise analysis. In addition to these anticipated features, the book breaks new ground by including the latest results on the highly efficient G-symplectic methods which compete strongly with the well-known symplectic Runge-Kutta methods for long-term integration of conservative mechanical systems.

This third edition of Numerical Methods for Ordinary Differential Equations will serve as a key text for senior undergraduate and graduate courses in numerical analysis, and is an essential resource for research workers in applied mathematics, physics and engineering.




Table of contents

Foreword xiii

Preface to the first edition xv

Preface to the second edition xix

Preface to the third edition xxi

1 Differential and Difference Equations 1

10 Differential Equation Problems 1

100 Introduction to differential equations 1

101 The Kepler problem 4

102 A problem arising from the method of lines 7

103 The simple pendulum 11

104 A chemical kinetics problem 14

105 The Van der Pol equation and limit cycles 16

106 The Lotka-Volterra problem and periodic orbits 18

107 The Euler equations of rigid body rotation 20

11 Differential Equation Theory 22

110 Existence and uniqueness of solutions 22

111 Linear systems of differential equations 24

112 Stiff differential equations 26

12 Further Evolutionary Problems 28

120 Many-body gravitational problems 28

121 Delay problems and discontinuous solutions 30

122 Problems evolving on a sphere 33

123 Further Hamiltonian problems 35

124 Further differential-algebraic problems 36

13 Difference Equation Problems 38

130 Introduction to difference equations 38

131 A linear problem 39

132 The Fibonacci difference equation 40

133 Three quadratic problems 40

134 Iterative solutions of a polynomial equation 41

135 The arithmetic-geometric mean 43

14 Difference Equation Theory 44

140 Linear difference equations 44

141 Constant coefficients 45

142 Powers of matrices 46

15 Location of Polynomial Zeros 50

150 Introduction 50

151 Left half-plane results 50

152 Unit disc results 52

Concluding remarks 53

2 Numerical Differential Equation Methods 55

20 The Euler Method 55

200 Introduction to the Euler method 55

201 Some numerical experiments 58

202 Calculations with stepsize control 61

203 Calculations with mildly stiff problems 65

204 Calculations with the implicit Euler method 68

21 Analysis of the Euler Method 70

210 Formulation of the Euler method 70

211 Local truncation error 71

212 Global truncation error 72

213 Convergence of the Euler method 73

214 Order of convergence 74

215 Asymptotic error formula 78

216 Stability characteristics 79

217 Local truncation error estimation 84

218 Rounding error 85

22 Generalizations of the Euler Method 90

220 Introduction 90

221 More computations in a step 90

222 Greater dependence on previous values 92

223 Use of higher derivatives 92

224 Multistep-multistage-multiderivative methods 94

225 Implicit methods 95

226 Local error estimates 96

23 Runge-Kutta Methods 97

230 Historical introduction 97

231 Second order methods 98

232 The coefficient tableau 98

233 Third order methods 99

234 Introduction to order conditions 100

235 Fourth order methods 101

236 Higher orders 103

237 Implicit Runge-Kutta methods 103

238 Stability characteristics 104

239 Numerical examples 108

24 Linear MultistepMethods 111

240 Historical introduction 111

241 Adams methods 111

242 General form of linear multistep methods 113

243 Consistency, stability and convergence 113

244 Predictor-corrector Adams methods 115

245 The Milne device 117

246 Starting methods 118

247 Numerical examples 119

25 Taylor Series Methods 120

250 Introduction to Taylor series methods 120

251 Manipulation of power series 121

252 An example of a Taylor series solution 122

253 Other methods using higher derivatives 123

254 The use of f derivatives 126

255 Further numerical examples 126

26 MultivalueMulitistage Methods 128

260 Historical introduction 128

261 Pseudo Runge-Kutta methods 128

262 Two-step Runge-Kutta methods 129

263 Generalized linear multistep methods 130

264 General linear methods 131

265 Numerical examples 133

27 Introduction to Implementation 135

270 Choice of method 135

271 Variable stepsize 136

272 Interpolation 138

273 Experiments with the Kepler problem 138

274 Experiments with a discontinuous problem 139

Concluding remarks 142

3 Runge-KuttaMethods 143

30 Preliminaries 143

300 Trees and rooted trees 143

301 Trees, forests and notations for trees 146

302 Centrality and centres 147

303 Enumeration of trees and unrooted trees 150

304 Functions on trees 153

305 Some combinatorial questions 155

306 Labelled trees and directed graphs 156

307 Differentiation 159

308 Taylor's theorem 161

31 Order Conditions 163

310 Elementary differentials 163

311 The Taylor expansion of the exact solution 166

312 Elementary weights 168

313 The Taylor expansion of the approximate solution 171

314 Independence of the elementary differentials 174

315 Conditions for order 174

316 Order conditions for scalar problems 175

317 Independence of elementary weights 178

318 Local truncation error 180

319 Global truncation error 181

32 Low Order ExplicitMethods 185

320 Methods of orders less than 4 185

321 Simplifying assumptions 186

322 Methods of order 4 189

323 New method