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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