Numerical Methods for Ordinary Differential Equations

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John Wiley & Sons, Aug 29, 2016 - Mathematics - 538 pages

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.

Key features:

?œ Presents a comprehensive and detailed study of the subject

?œ Covers both practical and theoretical aspects

?œ Includes widely accessible topics along with sophisticated and advanced details

?œ Offers a balance between traditional aspects and modern developments

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.

 

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Contents

Differential and Difference Equations
1
The Kepler problem
4
A problem arising from the method of lines
7
The simple pendulum
11
A chemical kinetics problem
14
The Van der Pol equation and limit cycles
16
The LotkaVolterra problem and periodic orbits
18
The Euler equations of rigid body rotation
20
Order stars
253
Order arrows and the Ehle barrier
256
ANstability
259
Nonlinear stability
262
BNstability of collocation methods
265
The V and W transformations
267
Implementable Implicit RungeKutta Methods
272
Diagonally implicit RungeKutta methods
273

Differential Equation Theory
22
Linear systems of differential equations
24
Stiff differential equations
26
Further Evolutionary Problems
28
Delay problems and discontinuous solutions
30
Problems evolving on a sphere
33
Further Hamiltonian problems
35
Further differentialalgebraic problems
36
Difference Equation Problems
38
A linear problem
39
The Fibonacci difference equation
40
Iterative solutions of a polynomial equation
41
The arithmeticgeometric mean
43
Difference Equation Theory
44
Constant coefficients
45
Powers of matrices
46
Location of Polynomial Zeros
50
Unit disc results
52
Concluding remarks
53
Numerical Differential Equation Methods
55
Some numerical experiments
58
Calculations with stepsize control
61
Calculations with mildly stiff problems
65
Calculations with the implicit Euler method
68
Analysis of the Euler Method
70
Local truncation error
71
Global truncation error
72
Convergence of the Euler method
73
Order of convergence
74
Asymptotic error formula
78
Stability characteristics
79
Local truncation error estimation
84
Rounding error
85
Generalizations of the Euler Method
90
Greater dependence on previous values
92
Multistepmultistagemultiderivative methods
94
Implicit methods
95
Local error estimates
96
RungeKutta Methods
97
Second order methods
98
Third order methods
99
Introduction to order conditions
100
Fourth order methods
101
Higher orders
103
Stability characteristics
104
Numerical examples
108
Linear Multistep Methods
111
General form of linear multistep methods
113
Predictorcorrector Adams methods
115
The Milne device
117
Starting methods
118
Numerical examples
119
Taylor Series Methods
120
Manipulation of power series
121
An example of a Taylor series solution
122
Other methods using higher derivatives
123
The use of f derivatives
126
Multivalue Mulitistage Methods
128
Twostep RungeKutta methods
129
Generalized linear multistep methods
130
General linear methods
131
Numerical examples
133
Introduction to Implementation
135
Variable stepsize
136
Interpolation
138
Experiments with a discontinuous problem
139
Concluding remarks
142
RungeKutta Methods
143
Trees forests and notations for trees
146
Centrality and centres
147
Enumeration of trees and unrooted trees
150
Functions on trees
153
Some combinatorial questions
155
Labelled trees and directed graphs
156
Differentiation
159
Taylors theorem
161
Order Conditions
163
The Taylor expansion of the exact solution
166
Elementary weights
168
The Taylor expansion of the approximate solution
171
Independence of the elementary differentials
174
Order conditions for scalar problems
175
Independence of elementary weights
178
Local truncation error
180
Global truncation error
181
Low Order Explicit Methods
185
Simplifying assumptions
186
Methods of order 4
189
New methods from old
195
Order barriers
200
Methods of order 5
204
Methods of order 6
206
Methods of order greater than 6
209
RungeKutta Methods with Error Estimates
211
Methods with builtin estimates
214
A class of errorestimating methods
215
The methods of Fehlberg
221
The methods of Verner
223
Implicit RungeKutta Methods
226
Solvability of implicit equations
227
Methods based on Gaussian quadrature
228
Reflected methods
233
Methods based on Radau and Lobatto quadrature
236
Stability of Implicit RungeKutta Methods
243
Criteria for Astability
244
Pade approximations to the exponential function
246
Astability of Gauss and related methods
252
The importance of high stage order
274
Singly implicit methods
278
Generalizations of singly implicit methods
283
Effective order and DESIRE methods
285
Implementation Issues
288
Acceptance and rejection of steps
290
Error per step versus error per unit step
291
Controltheoretic considerations
292
Solving the implicit equations
293
Algebraic Properties of RungeKutta Methods
296
Equivalence classes of RungeKutta methods
297
The group of RungeKutta tableaux
299
The RungeKutta group
302
A homomorphism between two groups
308
A generalization of G1
309
Some special elements of G
311
Some subgroups and quotient groups
314
An algebraic interpretation of effective order
316
Symplectic RungeKutta Methods
323
Hamiltonian mechanics and symplectic maps
324
Applications to variational problems
325
Examples of symplectic methods
326
Order conditions
327
Experiments with symplectic methods
328
Concluding remarks
331
Linear Multistep Methods
333
Starting methods
334
Convergence
335
Stability
336
Necessity of conditions for convergence
338
Sufficiency of conditions for convergence
339
The Order of Linear Multistep Methods
344
Derivation of methods
346
Backward difference methods
347
Errors and Error Growth
348
Further remarks on error growth
350
The underlying onestep method
352
Weakly stable methods
354
Variable stepsize
355
Stability Characteristics
357
Stability regions
359
Examples of the boundary locus method
360
An example of the Schur criterion
363
Stability of predictorcorrector methods
364
Order and Stability Barriers
367
Maximum order for a convergent kstep method
368
Order stars for linear multistep methods
371
Order arrows for linear multistep methods
373
Oneleg Methods and Gstability
375
The concept of Gstability
376
Transformations relating oneleg and linear multistep methods
379
Effective order interpretation
380
Implementation Issues
381
Representation of data
382
Variable stepsize for Nordsieck methods
385
Local error estimation
386
Concluding remarks
387
General Linear Methods
389
Transformations of methods
391
RungeKutta methods as general linear methods
392
Linear multistep methods as general linear methods
393
Some known unconventional methods
396
Some recently discovered general linear methods
398
Consistency Stability and Convergence
400
Covariance of methods
401
Definition of convergence
403
The necessity of stability
404
Stability and consistency imply convergence
406
The Stability of General Linear Methods
412
Methods with maximal stability order
413
Outline proof of the ButcherChipman conjecture
417
Nonlinear stability
419
Reducible linear multistep methods and Gstability
422
The Order of General Linear Methods
423
Local and global truncation errors
425
Algebraic analysis of order
426
An example of the algebraic approach to order
428
The underlying onestep method
429
Methods with RungeKutta stability
431
The types of DIMSIM methods
432
RungeKutta stability
435
Almost RungeKutta methods
438
Third order threestage ARK methods
441
Fourth order fourstage ARK methods
443
A fifth order fivestage method
446
Methods with Inherent RungeKutta Stability
448
Inherent RungeKutta stability
450
Conditions for zero spectral radius
452
Derivation of methods with IRK stability
454
Methods with property F
457
Some nonstiff methods
458
Some stiff methods
459
Scale and modify for stability
460
Scale and modify for error estimation
462
Gsymplectic methods
464
The control of parasitism
467
Order conditions
471
Two fourth order methods
474
Starters and finishers for sample methods
476
Simulations
480
Cohesiveness
481
The role of symmetry
487
Efficient starting
492
Concluding remarks
497
References
499
Index
509
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About the author (2016)

J.C Butcher, Emeritus Professor, University of Auckland, New Zealand

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