Numerical Methods for Ordinary Differential Equations

✍ Scribed by Butcher, J C

• Publisher: Wiley
• Year: 2016
• Tongue: English
• Leaves: 540
• Category: Library

✦ Synopsis

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 ofNumerical Methods for Ordinary Differential Equationswill 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

Cover......Page 1
Title Page......Page 5
Copyright......Page 6
Contents......Page 7
Foreword......Page 15
Preface to the first edition......Page 17
Preface to the second edition......Page 21
Preface to the third edition......Page 23
100 Introduction to differential equations......Page 27
101 The Kepler problem......Page 30
102 A problem arising from the method of lines......Page 33
103 The simple pendulum......Page 37
104 A chemical kinetics problem......Page 40
105 The Van der Pol equation and limit cycles......Page 42
106 The Lotka–Volterra problem and periodic orbits......Page 44
107 The Euler equations of rigid body rotation......Page 46
110 Existence and uniqueness of solutions......Page 48
111 Linear systems of differential equations......Page 50
112 Stiff differential equations......Page 52
120 Many-body gravitational problems......Page 54
121 Delay problems and discontinuous solutions......Page 56
122 Problems evolving on a sphere......Page 59
123 Further Hamiltonian problems......Page 61
124 Further differential-algebraic problems......Page 62
130 Introduction to difference equations......Page 64
131 A linear problem......Page 65
133 Three quadratic problems......Page 66
134 Iterative solutions of a polynomial equation......Page 67
135 The arithmetic-geometric mean......Page 69
140 Linear difference equations......Page 70
141 Constant coefficients......Page 71
142 Powers of matrices......Page 72
151 Left half-plane results......Page 76
152 Unit disc results......Page 78
Concluding remarks......Page 79
200 Introduction to the Euler method......Page 81
201 Some numerical experiments......Page 84
202 Calculations with stepsize control......Page 87
203 Calculations with mildly stiff problems......Page 91
204 Calculations with the implicit Euler method......Page 94
210 Formulation of the Euler method......Page 96
211 Local truncation error......Page 97
212 Global truncation error......Page 98
213 Convergence of the Euler method......Page 99
214 Order of convergence......Page 100
215 Asymptotic error formula......Page 104
216 Stability characteristics......Page 105
217 Local truncation error estimation......Page 110
218 Rounding error......Page 111
221 More computations in a step......Page 116
223 Use of higher derivatives......Page 118
224 Multistep–multistage–multiderivative methods......Page 120
225 Implicit methods......Page 121
226 Local error estimates......Page 122
230 Historical introduction......Page 123
232 The coefficient tableau......Page 124
233 Third order methods......Page 125
234 Introduction to order conditions......Page 126
235 Fourth order methods......Page 127
237 Implicit Runge–Kutta methods......Page 129
238 Stability characteristics......Page 130
239 Numerical examples......Page 134
241 Adams methods......Page 137
243 Consistency, stability and convergence......Page 139
244 Predictor–corrector Adams methods......Page 141
245 The Milne device......Page 143
246 Starting methods......Page 144
247 Numerical examples......Page 145
250 Introduction to Taylor series methods......Page 146
251 Manipulation of power series......Page 147
252 An example of a Taylor series solution......Page 148
253 Other methods using higher derivatives......Page 149
255 Further numerical examples......Page 152
261 Pseudo Runge–Kutta methods......Page 154
262 Two-step Runge–Kutta methods......Page 155
263 Generalized linear multistep methods......Page 156
264 General linear methods......Page 157
265 Numerical examples......Page 159
270 Choice of method......Page 161
271 Variable stepsize......Page 162
273 Experiments with the Kepler problem......Page 164
274 Experiments with a discontinuous problem......Page 165
Concluding remarks......Page 168
300 Trees and rooted trees......Page 169
301 Trees, forests and notations for trees......Page 172
302 Centrality and centres......Page 173
303 Enumeration of trees and unrooted trees......Page 176
304 Functions on trees......Page 179
305 Some combinatorial questions......Page 181
306 Labelled trees and directed graphs......Page 182
307 Differentiation......Page 185
308 Taylor’s theorem......Page 187
310 Elementary differentials......Page 189
311 The Taylor expansion of the exact solution......Page 192
312 Elementary weights......Page 194
313 The Taylor expansion of the approximate solution......Page 197
315 Conditions for order......Page 200
316 Order conditions for scalar problems......Page 201
317 Independence of elementary weights......Page 204
318 Local truncation error......Page 206
319 Global truncation error......Page 207
320 Methods of orders less than 4......Page 211
321 Simplifying assumptions......Page 212
322 Methods of order 4......Page 215
323 New methods from old......Page 221
324 Order barriers......Page 226
325 Methods of order 5......Page 230
326 Methods of order 6......Page 232
327 Methods of order greater than 6......Page 235
331 Richardson error estimates......Page 237
332 Methods with built-in estimates......Page 240
333 A class of error-estimating methods......Page 241
334 The methods of Fehlberg......Page 247
336 The methods of Dormand and Prince......Page 249
340 Introduction......Page 252
341 Solvability of implicit equations......Page 253
342 Methods based on Gaussian quadrature......Page 254
343 Reflected methods......Page 259
344 Methods based on Radau and Lobatto quadrature......Page 262
350 A-stability, A(a)-stability and L-stability......Page 269
351 Criteria for A-stability......Page 270
352 Pade approximations to the exponential function......Page 271
353 A-stability of Gauss and related methods......Page 278
354 Order stars......Page 279
355 Order arrows and the Ehle barrier......Page 282
356 AN-stability......Page 285
357 Non-linear stability......Page 288
358 BN-stability of collocation methods......Page 291
359 The V and W transformations......Page 293
360 Implementation of implicit Runge–Kutta methods......Page 298
361 Diagonally implicit Runge–Kutta methods......Page 299
362 The importance of high stage order......Page 300
363 Singly implicit methods......Page 304
364 Generalizations of singly implicit methods......Page 309
365 Effective order and DESIRE methods......Page 311
371 Optimal sequences......Page 314
372 Acceptance and rejection of steps......Page 316
373 Error per step versus error per unit step......Page 317
374 Control-theoretic considerations......Page 318
375 Solving the implicit equations......Page 319
380 Motivation......Page 322
381 Equivalence classes of Runge–Kutta methods......Page 323
382 The group of Runge–Kutta tableaux......Page 325
383 The Runge–Kutta group......Page 328
384 A homomorphism between two groups......Page 334
385 A generalization of G1......Page 335
386 Some special elements of G......Page 337
387 Some subgroups and quotient groups......Page 340
388 An algebraic interpretation of effective order......Page 342
390 Maintaining quadratic invariants......Page 349
391 Hamiltonian mechanics and symplectic maps......Page 350
392 Applications to variational problems......Page 351
393 Examples of symplectic methods......Page 352
394 Order conditions......Page 353
395 Experiments with symplectic methods......Page 354
Concluding remarks......Page 357
400 Fundamentals......Page 359
401 Starting methods......Page 360
402 Convergence......Page 361
404 Consistency......Page 362
405 Necessity of conditions for convergence......Page 364
406 Sufficiency of conditions for convergence......Page 365
410 Criteria for order......Page 370
411 Derivation of methods......Page 372
412 Backward difference methods......Page 373
420 Introduction......Page 374
421 Further remarks on error growth......Page 376
422 The underlying one-step method......Page 378
423 Weakly stable methods......Page 380
424 Variable stepsize......Page 381
430 Introduction......Page 383
431 Stability regions......Page 385
432 Examples of the boundary locus method......Page 386
433 An example of the Schur criterion......Page 389
434 Stability of predictor–corrector methods......Page 390
440 Survey of barrier results......Page 393
441 Maximum order for a convergent k-step method......Page 394
442 Order stars for linear multistep methods......Page 397
443 Order arrows for linear multistep methods......Page 399
450 The one-leg counterpart to a linear multistep method......Page 401
451 The concept of G-stability......Page 402
452 Transformations relating one-leg and linear multistep methods......Page 405
454 Concluding remarks on G-stability......Page 406
460 Survey of implementation considerations......Page 407
461 Representation of data......Page 408
462 Variable stepsize for Nordsieck methods......Page 411
463 Local error estimation......Page 412
Concluding remarks......Page 413
500 Multivalue–multistage methods......Page 415
501 Transformations of methods......Page 417
502 Runge–Kutta methods as general linear methods......Page 418
503 Linear multistep methods as general linear methods......Page 419
504 Some known unconventional methods......Page 422
505 Some recently discovered general linear methods......Page 424
510 Definitions of consistency and stability......Page 426
511 Covariance of methods......Page 427
512 Definition of convergence......Page 429
514 The necessity of consistency......Page 430
515 Stability and consistency imply convergence......Page 432
520 Introduction......Page 438
521 Methods with maximal stability order......Page 439
522 Outline proof of the Butcher–Chipman conjecture......Page 443
523 Non-linear stability......Page 445
524 Reducible linear multistep methods and G-stability......Page 448
530 Possible definitions of order......Page 449
531 Local and global truncation errors......Page 451
532 Algebraic analysis of order......Page 452
533 An example of the algebraic approach to order......Page 454
534 The underlying one-step method......Page 455
540 Design criteria for general linear methods......Page 457
541 The types of DIMSIM methods......Page 458
542 Runge–Kutta stability......Page 461
543 Almost Runge–Kutta methods......Page 464
544 Third order, three-stage ARK methods......Page 467
545 Fourth order, four-stage ARK methods......Page 469
547 ARK methods for stiff problems......Page 472
550 Doubly companion matrices......Page 474
551 Inherent Runge–Kutta stability......Page 476
552 Conditions for zero spectral radius......Page 478
553 Derivation of methods with IRK stability......Page 480
554 Methods with property F......Page 483
555 Some non-stiff methods......Page 484
556 Some stiff methods......Page 485
557 Scale and modify for stability......Page 486
558 Scale and modify for error estimation......Page 488
560 Introduction......Page 490
561 The control of parasitism......Page 493
562 Order conditions......Page 497
563 Two fourth order methods......Page 500
564 Starters and finishers for sample methods......Page 502
565 Simulations......Page 506
566 Cohesiveness......Page 507
567 The role of symmetry......Page 513
568 Efficient starting......Page 518
Concluding remarks......Page 523
References......Page 525
Index......Page 535
EULA......Page 540